Parametric Analysis of Resilient Design of Steel Truss Bridges
By
Stephanie Borchers
Bachelor of Engineering in Civil Engineering
The Cooper Union for the Advancement of Science and Art, 2014
Submitted to the Department of Civil and Environmental Engineering in Partial Fulfillment of
the Requirements for the Degree of
ARGHJS
Master of Engineering in Civil and Environmental Engineering
at the
Massachusetts Institute of Technology
MASSACHUSETTS INSTITUTE
OF [ECHNOLOLGY
June 2015
JUL 02 2015
LIBRARIES
C 2015 Stephanie Borchers. All Rights Reserved.
The author hereby grants to MIT permission to reproduce and to distribute publicly paper and
electronic copies of this thesis document in whole or in part in any medium now known or
hereafter created.
Signature of Author:
Signature redacted
Deparient of tivil and Environmental Engineering
May 18, 2015
Certified By:
Signature redacted
4ierre Ghisbain
Lecturer of Civil and Environmental Engineering
Thesis Supervisor
I
4
Accepted By:
Signature redacted
idi Nepf
Donald and Martha Harleman Professor of Civil and Environmental E gineering
Chair, Departmental Committee for Graduate Students
Parametric Analysis of Resilient Design of Steel Truss Bridges
By
Stephanie Borchers
Submitted to the Department of Civil and Environmental Engineering on May 18, 2015 in Partial
Fulfillment of the Requirements for the Degree of Master of Engineering in Civil and
Environmental Engineering
Abstract
Designing structures to be resilient to extreme loads has become a topic of interest in recent
years, which has been triggered by the progressive collapse of structures in the past. Structural
failure due to the lack of resilient design has been particularly prevalent in bridges. The failures
have been results of a variety of factors that the bridges have been subjected to. The objective of
preventing the occurrence of future collapses has encouraged further research into the design of
resilient structures.
Two main methods to design for resilience have been implemented in this thesis. These methods
include the incorporation of robustness or redundancy into the bridge design. Each method is
advantageous over the other in certain circumstances. These methods are both based on linear
static analysis procedures.
A series of 2D truss bridge models with varying parameters have been analyzed for their
performance in damaged states. The damage incurred by the bridges include the removal of a
pier and the removal of bridge members. The results of this investigation conclude that the cost
of designing a bridge to be resilient is relatively low in comparison to the overall cost of the
bridge. Robust bridge designs are generally more effective for bridges with longer spans,
whereas designs with redundancy are better suited for shorter spans. As the amount of structural
damage that is incurred by a bridge increases, the more redundancy should be built into the
structure. These results were shared by all three of the truss topologies that were explored.
Thesis Supervisor: Pierre Ghisbain
Title: Lecturer, Department of Civil and Environmental Engineering
3
Acknowledgements
First, I would like to thank Pierre Ghisbain for his guidance with this thesis and his constant
availability. His devotion to his students is unparalleled, and is greatly appreciated by the entire
class. I would also like to thank the rest of the M. Eng. faculty and administration for their
commitment throughout the year.
Secondly, I would like to thank my family for their love and support, and their utmost confidence
in me. Their constant encouragement has motivated me to continue to strive during this difficult
year.
I would like to thank Natalia for her friendship and support, and for providing me with countless,
much needed study breaks. She has kept me sane throughout this stressful year, and it has been a
pleasure being in this program with her.
Lastly, I would like to thank my MEng classmates for an unforgettable year. This incredible
group of people has influenced me in a positive way, and has provided me with friendships from
all over the world. I couldn't have asked for a better class to complete this program with.
5
Table of Contents
1 Background ................................................................................................................................
12
Intro ................................................................................................................................
12
1.1.1 Robustness....................................................................................................................
12
1.1.2 Redundancy..................................................................................................................
12
1.2 Causes of Bridge Collapse ...............................................................................................
13
1.3 Exam ples of Bridge Collapse...........................................................................................
14
1.3.1 1-5 Skagit River ............................................................................................................
14
1.3.2 1-40 Bridge....................................................................................................................
15
1.3.3 Skieggestad Bridge....................................................................................................
16
1.1
1.4 Objective .............................................................................................................................
17
2 A nalysis Procedure ....................................................................................................................
17
2.1 Fram e M odel (SAP2000).................................................................................................
18
2.1.1 U niform Linear Static Load..........................................................................................
18
2.1.2 Rem oval of Pier............................................................................................................
22
2.1.3 Rem oval of M embers ...............................................................................................
23
2.2 Truss M odel (Matlab) ......................................................................................................
24
2.2.1 Point Loads...................................................................................................................
24
2.2.2 Rem oval of Pier, Full and Half Load ........................................................................
24
2.2.3 Rem oval of 1 M ember, Full and Half Load .............................................................
24
2.3 Com parison of Analysis Methods....................................................................................
3 Results........................................................................................................................................
25
26
3.1 Specifications ......................................................................................................................
26
3.2 Sum m ary .............................................................................................................................
28
3.3 Analysis of W arren Truss using Fram e M odel ...............................................................
31
3.3.1 W arren Truss after Support Rem oval.........................................................................
31
3.3.2 W arren Truss after M em ber Rem oval......................................................................
31
3.4 Analysis of K -Truss using Fram e M odel........................................................................
35
3.4.1 K -Truss after Support Rem oval..................................................................................
35
3.4.2 K -Truss after M em ber Rem oval...............................................................................
36
3.5 Analysis of Pratt Truss using Fram e M odel....................................................................
38
3.5.1 Pratt-Truss after M ember Rem oval...........................................................................
38
7
3.6 Analysis of W arren Truss using Truss M odel..................................................................
40
3.6.1 Com parison to Fram e Model....................................................................................
40
3.6.2 W arren Truss under Reduced Load...........................................................................
41
42
3.7 Cost Analysis ......................................................................................................................
3.7.1 Cost of W ithstanding Rem oval of Pier.......................................................................
42
3.7.2 Cost of W ithstanding Rem oval of One M em ber......................................................
43
3.7.3 Cost of Withstanding Removal of Two Members.........................................
43
3.7.4 Cost of W ithstanding Rem oval of Three M embers....................................................
44
3.7.5 Discussion on Truss Model
3.7.6 Additional Notes on Cost
..............
..............................
...... 45
.......................................................................................
47
.......................................
... 47
3.8.1 Vessel Collision ..................................................................................................
48
3.8 Probability of Collapse
.... ......................................
3.8.2 Vehicle Collision..........................................................................................
..... 52
4 Conclusions.......( .................................................................................................................
.53
4.1 Discussion............................................................................................................................
53
4.2 Future Work/Im pact ............................................................................................................
54
5 References..................................................................................................................................
55
6 Appendix....................................................................................................................................
56
6.1 Robust and Redundant Bridge M odels after Dam age......................................................
8
56
List of Figures
Figure 1: Causes of Bridge Collapse since 1900 ......................................................................
Figure 2: Partial Collapse of 1-5 Skagit River Bridge ...............................................................
Figure 3: Partial Collapse of 1-40 Bridge .................................................................................
Figure 4: Partial Collapse of Skieggestad Bridge......................................................................
Figure 5: Importing Frame Sections ...........................................................................................
Figure 6: Selecting Frame Property File....................................................................................
Figure 7: Adding Frame Sections .............................................................................................
Figure 8: Creating Auto Select Sections....................................................................................
Figure 9: Assigning Frame Properties ......................................................................................
Figure 10: Bridge Model with Auto Sections Assigned ...........................................................
Figure 11: Load Pattern Definition...........................................................................................
Figure 12: Load Case Definition ...............................................................................................
Figure 13: Load Combination Definition .................................................................................
Figure 14: Linear Static Analysis Definition.............................................................................
Figure 15: Steel Frame Design Preferences................................................................................
Figure 16: Analysis versus Design Section Check ....................................................................
Figure 17: Steel Member Check ...............................................................................................
Figure 18: Warren Truss 1.5:1 after running the Design/Check of Structure............................
Figure 19: Warren Truss 1.5:1 after a support was removed.....................................................
Figure 20: Warren Truss 1.5:1 after resizing of members.........................................................
Figure 21: Dimensions of 0.5:1 Warren Truss ..........................................................................
Figure 22: Dimensions of 0.75:1 Warren Truss ........................................................................
Figure 23: Dimensions of 1:1 Warren Truss .............................................................................
Figure 24: Dimensions of 1.25:1 Warren Truss ........................................................................
Figure 25: Dimensions of 1.5:1 Warren Truss ..........................................................................
Figure 26: Dimensions of 1.75:1 Warren Truss ........................................................................
Figure 27: Dimensions of 1:1 K-Truss ......................................................................................
Figure 28: Dimensions of 1.5:1 K-Truss ....................................................................................
Figure 29: Dimensions of 2:1 K-Truss ......................................................................................
Figure 30: Dimensions of 1:1 Pratt Truss..................................................................................
Figure 31: Dimensions of 1.25:1 Pratt Truss.............................................................................
Figure 32: Dimensions of 1.5:1 Pratt Truss...............................................................................
Figure 33: Sample Robust and Redundant Designs for All Truss Topologies..........................
Figure 34: Weight of Undamaged Bridge vs C:S Span Ratio for All Truss Models.................
Figure 35: Weight Ratio of Undamaged Bridge vs. C:S Span Ratio for All Truss Models.........
Figure 36: Weight Ratio of Bridge vs. No. of Elements Removed for Warren Truss Models.....
Figure 37: Weight Ratio of Bridge vs. No. of Elements Removed for Warren 2 Truss Models..
Figure 38: Weight Ratio of New Bridge to Original Bridge vs. Number of Elements Removed
W arren Truss M odels....................................................................................................................
Figure 39: Weight Ratio of New Bridge to Original Bridge vs. Number of Elements Removed
W arren2 Truss M odels..................................................................................................................
9
13
14
15
16
19
19
20
20
20
20
21
21
21
21
21
22
22
25
25
26
27
27
27
27
27
27
27
27
27
28
28
28
28
30
30
33
34
for
34
for
35
Figure 40: Weight Ratio of Bridge vs. Number of Elements Removed for K-Truss Models ...... 37
Figure 41: Weight Ratio of New Bridge to Original Bridge vs. Number of Elements Removed for
K -T russ M odels ............................................................................................................................
37
Figure 42: Weight Ratio of Bridge vs. Number of Elements Removed for Pratt Truss Models.. 39
Figure 43: Weight Ratio of New Bridge to Original Bridge vs. Number of Elements Removed for
Pratt Truss M odels ........................................................................................................................
40
Figure 44: Weight Ratio of New Bridge to Original Bridge vs. C:S Span Ratio after the Removal
o f a S upp ort...................................................................................................................................
42
Figure 45: Weight Ratio of New Bridge to Original Bridge vs. C:S Span Ratio after the Removal
of O ne M emb er.............................................................................................................................
43
Figure 46: Weight Ratio of New Bridge to Original Bridge vs. C:S Span Ratio after the Removal
of T w o Mem bers...........................................................................................................................
44
Figure 47: Weight Ratio of New Bridge to Original Bridge vs. C:S Span Ratio after the Removal
of T hree M em bers.........................................................................................................................
45
Figure 48: F.R. vs. C:S Span Ratio for Warren Truss Models after the Removal of a Support... 46
Figure 49: F.R. vs. C:S Span Ratio for Warren Truss Models after the Removal of a Member.. 46
Figure 50: Barge Impact Force, Deformation Energy, and Damage Length Data (AASHTO) ... 49
Figure 51: Probability of Collapse Distribution (AASHTO) ...................................................
51
Figure 52: Robust and Redundant 0.5:1 Warren Truss after Support Removal....................... 56
Figure 53: Robust and Redundant 0.5:1 Warren Truss after Member Removal ....................... 56
Figure 54: Robust and Redundant 0.75:1 Warren Truss after Support Removal...................... 56
Figure 55: Robust and Redundant 0.75:1 Warren Truss after Member Removal..................... 56
Figure 56: Robust and Redundant 1:1 Warren Truss after Support Removal........................... 56
Figure 57: Robust and Redundant 1:1 Warren Truss after Member Removal ......................... 57
Figure 58: Robust and Redundant 1.25:1 Warren Truss after Support Removal...................... 57
Figure 59: Robust and Redundant 1.25:1 Warren Truss after Member Removal ..................... 57
Figure 60: Robust and Redundant 1.5:1 Warren Truss after Support Removal........................ 57
Figure 61: Robust and Redundant 1.5:1 Warren Truss after Member Removal ....................... 57
Figure 62: Robust and Redundant 1.75:1 Warren Truss after Support Removal...................... 57
Figure 63: Robust and Redundant 1.75:1 Warren Truss after Member Removal..................... 58
Figure 64: Robust and Redundant 1:1 K-Truss after Support Removal....................................
58
Figure 65 Robust and Redundant 1:1 K-Truss after Member Removal....................................
58
Figure 66: Robust and Redundant 1.5:1 K-Truss after Support Removal................................. 58
Figure 67: Robust and Redundant 1.5:1 K-Truss after Member Removal............................... 58
Figure 68: Robust and Redundant 2:1 K-Truss after Support Removal....................................
58
Figure 69: Robust and Redundant 2:1 K-Truss after Member Removal...................................
59
Figure 70: Robust and Redundant 1:1 Pratt Truss after Support Removal ............................... 59
Figure 71: Robust and Redundant 1:1 Pratt Truss after Member Removal............................... 59
Figure 72: Robust and Redundant 1.25:1 Pratt Truss after Support Removal .......................... 59
Figure 73: Robust and Redundant 1.25:1 Pratt Truss after Member Removal.......................... 59
Figure 74: Robust and Redundant 1.5:1 Pratt Truss after Support Removal ........................... 59
Figure 75: Robust and Redundant 1.5:1 Pratt Truss after Member Removal........................... 60
10
List of Tables
Table 1: Original Weights of Warren Truss Bridge Models in Kips........................................
29
Table 2: Original Weights of K-Truss Bridge Models in Kips..................................................
29
Table 3: Original Weights of Pratt Truss Bridge Models in Kips .............................................
29
Table 4: Weight Ratios of 0.5:1 Warren Truss under Parametric Conditions........................... 32
Table 5: Weight Ratios of 0.75:1 Warren Truss under Parametric Conditions......................... 32
Table 6: Weight Ratios of 1:1 Warren Truss under Parametric Conditions............................. 32
Table 7: Weight Ratios of 1.25:1 Warren Truss under Parametric Conditions......................... 33
Table 8: Weight Ratios of 1.5:1 Warren Truss under Parametric Conditions........................... 33
Table 9: Weight Ratios of 1.75:1 Warren Truss under Parametric Conditions......................... 33
Table 10: Weight Ratios of 1:1 K-Truss under Parametric Conditions.................................... 36
Table 11: Weight Ratios of 1.5:1 K-Truss under Parametric Conditions.................................. 36
Table 12: Weight Ratios of 2:1 K-Truss under Parametric Conditions.................................... 36
Table 13: Weight Ratios of 1:1 Pratt Truss under Parametric Conditions ............................... 38
Table 14: Weight Ratios of 1.25:1 Pratt Truss under Parametric Conditions ........................... 38
Table 15: Weight Ratios of 1.5:1 Pratt Truss under Parametric Conditions ............................. 39
Table 16: Force Ratios of 1:1 Warren Truss under Full and Reduced Loads .......................... 41
Table 17: Force Ratios of 1.25:1 Warren Truss under Full and Reduced Loads ...................... 41
Table 18: Force Ratios of 1.5:1 Warren Truss under Full and Reduced Loads ........................ 42
Table 19: Annually Averaged Vessel Traffic Data for New St. George Island Bridge ............ 50
Table 20: Probabilty of Collapse of Bridge Pier due to Vessel Collision based on Traffic Data for
51
N ew St. G eorge Island Bridge ....................................................................................................
Table 21: Typical Values of the Annual Frequency for a Bridge Pier to be Hit by a Heavy
52
V ehicle (A A SH T O 2012).............................................................................................................
11
1 Background
1.1 Intro
Bridges are subject to a variety of factors during their lifetimes, which could lead to
structural failure. Those factors include boat impacts, earthquakes, severe wind loads, floods,
material defects, and flaws in design. Depending on the bridge design and location, each of
these factors has a probability of occurrence. Based on these probabilities, it may be possible to
predict the likelihood of the bridge's collapse.
There are several reports regarding the analysis of collapsed bridges and their causes.
For instance, the report on the 1-35W steel truss bridge in Minneapolis, Minnesota clarifies that
the dead load of the structure had increased due to repair and slab reinforcement. Additionally,
the thickness of the gusset plate that failed was half of its design value. These factors
contributed to the bridge's collapse. Similar investigation studies have been conducted on
existing structures, particularly bridges that are aging or deteriorating (Miyachia et al., 2012).
The disastrous structural collapses that have resulted in the past from local failures of
critical members have emphasized the significance of designing structures to be resilient. This
notion of designing structures to withstand extraordinary loads was first brought about with the
partial collapse of London's Ronan Point Apartment Tower in 1968. The structure's lack of
alternative load paths prevented the redistribution of forces after an internal explosion occurred.
To prevent future occurrences of collapse, engineers have been researching methods to design
structures to be resilient. Two methods of achieving resilience are by incorporating structural
redundancy and robustness in bridge design.
1.1.1 Robustness
In this thesis, robustness is defined as the ability of a structure to continue to carry load
after being brought to a damaged state. This means the structure will continue to stand after an
event damages a part of the structure. Essentially, the damage that is caused to one part of the
structure is not propagated to the rest of the structure.
1.1.2 Redundancy
Redundancy is defined as the ability of a structure to continue to carry load after the
failure of a single element. The forces that could no longer be taken by the removed element
12
would then be redistributed to the surrounding structure. Redundant designs generally
incorporate structural members that are not strictly necessary for the design loads, but are
essential in an extreme event.
1.2 Causes of Bridge Collapse
On record, there have been a total of 151 major bridge failures in the world since 1900.
Of these, 22 were due to boat impacts, beginning with the Portage Canal Swing Bridge in 1905.
The bridge was struck by a steamer, causing the bridge span to topple over into the canal. The
damage of structural elements by train crashes led to 7 bridge collapses since 1900, which have
typically resulted from impacts on structural elements by derailed trains. Structural damage by
heavy vehicles resulted in 5 collapses. Among the other bridge collapses, accidents during
construction was the most probable cause of bridge failure, at 32 collapses. These were followed
by odd occurrences and unforeseen loads, floods and scour, flaws in the structural design, poor
maintenance, and earthquakes. Some of the bridge collapses were not caused by a single event,
but were a result of more than one factor over the lifetime of the bridge. The figure below shows
the breakdown of the causes of bridge collapse over the past century.
Heavy Vehicles +
Train Crash
11%
Earthquake
2%
Fire
3%
Boat Impact
Design Flaws
14%
9%
Manufacturing
Defects
6%
Flood/Scour
17%
Construction
Accidents
28%
Figure 1: Causes of Bridge Collapse since 1900
13
1.3 Examples of Bridge Collapse
1.3.1 1-5 Skagit River
On May 23, 2013, the 1-5 Skagit River Bridge collapsed in Mount Vernon, Washington.
The bridge was a through-truss bridge that was built in 1955 to connect the cities of Mount
Vernon and Burlington. The collapse resulted in three severe injuries. It was caused by an
oversize load that struck several sway struts, and indirectly damaged the compression chords of
the overhead steel frame. This led to an immediate collapse of one of the bridge's four spans.
The bridge's design did not have redundant structural members to protect the structure in
the event of a failure of one of the bridge's support members. Because the bridge's spans were
structurally independent, only one of the spans collapsed while the other three remained intact.
The piers below the deck remained unaffected.
Because the bridge provided a vital route between Seattle and Vancouver, replacing the
collapsed span was urgent. While the bridge was being repaired, two temporary bridges were
constructed to provide alternative routes.
This collapse resulted from the lack of redistribution of forces to the rest of the structure
after the event. The collapse, injuries, and the costs associated with constructing the two
temporary bridges and repairing the original bridge could have been averted if the original design
provided more redundancy to the structure. In 2013, work had begun to prevent a similar failure
from occurring to the remaining three bridge spans (Welch, 2013).
Figure 2: Partial Collapse of 1-5 Skagit River Bridge
Source: Dennis Bratland
14
1.3.2 1-40 Bridge
In Webbers Falls, Oklahoma, the 1-40 Bridge was struck by two barges on May 26, 2002.
The captain of the towboat Robert Y. Love had fallen asleep while passing below the bridge. He
lost control of the barges being towed, allowing them to collide with one of the bridge's piers
(Hancock, 2002). As a result, a large section of the bridge plummeted into the Arkansas River.
Fourteen people were killed, along with eleven people that were injured. The bridge was
repaired and reopened just two months after the collapse, despite the six months that it was
expected to take (Hopkins, 2002).
The 1-40 Bridge was designed in sections, and the impact to one pier only affected the
nearby bridge deck. The forces from the impact were not redistributed to the entire structure,
and all of the load was taken by the single bridge pier. The damaged bridge pier was struck with
an impact of almost double its design capacity. One method to prevent future collapse of the
bridge from a similar event would be to increase the capacity of the bridge piers. This would
ensure that the pier would be able to sustain a larger impact before experiencing damage. The
installation of pier protection systems could also mitigate damage to the pier, by creating
protective barriers that would work to either deflect vessels or absorb their impact. Another
method would be to incorporate redundancy into the design the bridge, so that the loads would
be redistributed in the case of the collapse of a single pier (Assis, 2006).
Figure 3: Partial Collapse of 1-40 Bridge
Source: McLaren Engineering Group
15
1.3.3 Skieggestad Bridge
More recently, the Skieggestad Bridge in Holmestrand, Norway underwent a partial
collapse on February 2, 2015. It was reported that a landslide beneath the bridge caused one of
the bridge piers to give way, causing the entire bridge to sag. The collapse was due to the poor
soil conditions around the bridge's foundations, which consisted of quick clay. This collapse has
prompted investigations of similar structures in the country. The presence of quick clay beneath
Norwegian bridge foundations is not unusual, and therefore many bridges are at a similar risk.
Based on the investigations, the Norwegian Public Roads Association (NPRA) is taking
measures to prevent future collapses from occurring (Norwegian bridge collapse, 2015).
Figure 4: Partial Collapse of Skieggestad Bridge
Source: News in English
16
1.4 Objective
Particular bridges have been studied in detail for their redundancy and ability to
withstand certain events. However, there has not been a general study to quantify the cost of
designing bridges to be resilient. Therefore, the objective of this thesis is to evaluate bridge
design for resilience under different extreme events, while taking into account cost efficiency.
The bridge would be evaluated for robustness and redundancy by varying its properties. This
parametric analysis varies the span and topology of the bridge, and investigates the probabilities
that the bridge would incur any damage.
2 Analysis Procedure
There were two different analysis procedures that were carried out to design truss bridges
for resiliency. Each procedure implemented a finite element analysis to assess different bridge
designs under various conditions. The first procedure consisted of frame models, which was
followed by a truss model procedure to be utilized for comparison.
A series of 2 dimensional bridge models were developed. Each bridge has three spans,
which are supported on two central pin supports and two end roller supports. The parameters
that were chosen to vary are the central span, the bridge topology, and the number of members
for each bridge. The depth of the bridges were held constant. The bridge types analyzed in this
study are the Warren truss, the K-truss, and the Pratt truss. The geometries of each bridge
analyzed are shown in figures Figure 21 through Figure 32.
The following load combinations were applied:
1.4 DL
1.2 DL + 1.6 LL
where DL is the dead load, and LL is the live load. The dead load to be applied to the deck of
the structure was assumed to be 100 psf. A 30' width was assumed for the 2D models,
producing a 1.5 kip/ft uniform dead load on one side of the steel structure. The live load is
dependent on the occupancy of the bridge during an event, which was assumed at a maximum of
17
75 psf. This corresponds to a liner load of 1.13 kip/ft on one side of the structure. Using the
load combinations, the maximum uniform load to be applied is 3.6 kip/ft.
2.1 Frame Model (SAP2000)
SAP2000 was used to conduct a finite element analysis of the 2D bridge models. Several
assumptions were made when modeling the bridge frames, to ensure the consistency of the
models and to simplify the analysis. The 2D frames were considered to be moment connected.
2.1.1 Uniform Linear Static Load
The individual members of the bridge models were initially assigned to a series of
member sizes from a specified list, using the Auto Select List function in SAP2000. For the
purpose of this study, the list was specified to include sizes ranging from W12x14 to WI2x336.
A uniform factored live and dead load was then applied to the deck of each bridge model. A
linear static analysis was run for each model, assuming each member to be the average truss size
from the Auto Select List. After running the analysis, the structure was designed and optimized
using the Steel Design/Check of Structure function. The structure was checked for stress
capacities, bending, shear, and deflection. The detailed procedure is as follows:
1. Build a 2D bridge model in SAP2000.
2. Import new frame section properties. Choose I/Wide Flange as shown in Figure 5 below.
Select from Sections8 as shown in Figure 6, and select all W12 sections from W12x14 to
W12x336.
3. Add new property to the Auto Select List, and choose the W12 sections that were previously
added. See Figure 7 and Figure 8.
4. Assign the Auto Section property to all frame elements, as shown in Figure 9. The Warren
Truss 1.5:1 bridge with all automatically-selected sections is shown in Figure 10.
5. Define the Live and Dead load patterns each with a multiplier of 1, as shown in Figure 11.
6. Define a load case with a static load case type and a linear analysis type, as shown in Figure
12.
7. Define the load combinations by choosing Add Default Design Combos, from Figure 13.
18
8. Run the Linear Static analysis case, as seen in Figure 14.
9. Set the Steel Frame Design Preferences according to the AISC360-10 Design Code, and make
sure that the values match Figure 15 below.
10. Start Steel Design/Check of the Structure, and Verify the Analysis versus the Design Section.
If any of the sections differ as in Figure 16, repeat step 8 and this step until all sections match.
Then verify that all members passed, as in Figure 17.
11. Record the total structural weight of the bridge by choosing Display > Show Tables > Model
Definition > Other Definitions > Group Data > Masses and Weights.
1 ." Computen and Stnuct-
S~
IL"
fwf"o"
-rane
-
SAP2= 17
Eval
Now fotdtr
~
L
#
T
cIcWos3
swm
on1
PON
Twill
Chbese.pro
ChnsoGS0M.po
T PC
SA360 Ofle ott
CTSC9Ph
ctSUcgro
Des~opOSC10,ro
OIhSoft~svtedMmn.gemeont.CotrolLb...
DihSoftPjectManagnentframewo&k.
) 4M~K
EUROPRO
Sp.wn
Indiaonpro
3Videas
01.0
SECTONSPRO
Windows (C
RECVER
Q.
%IONSRO
Fk nwne SECTIONSLPRO
conclo
Figure 5: Importing Frame Sections
xOenk
aPiM
I-
-~
Figure 6: Selecting Frame Property File
19
v
ProputyFies (*pro)
Cline
'I
~
AuM
SOCM. NM
Aft
SWAn
Tu
I"
$*Coww
T
14
?j"
c*.am( kd.msbeo
LMt of Soclus
AuIe
S*laWiW
W12A14-A
W12X14-A
W12X19-A
____
WIZX3S-AI
W12X30A
WIZX35-4
stIXA-A
sw"3Ooaft
04
Cisc.
t
cFia
Figure 8: Creating Auto Select Sections
Figure 7: Adding Frame Sections
cft to
iMot Now Papty..
AcNeWPropertV.
TnMa
14
W1ru12
W12X14
W12X16
Uar*yishow Pnpwry.
W12X22
W12X26
W12)(30
Wi2X35
WIZX4
Domee
Praperty
W12X*4
71~71
Figure 9: Assigning Frame Properties
WI2X79
W2X791
Truss12j
fTus12)
W12X79
W12X9
rfuss12i
Trum12)
W12X79
W12X79
(Tis12i
(Tirkss2
W12X?9 Truss12
W12X79
tTussi)
W1X79f TIsS121
WIXgt
ruSS12
W12X79
Trus512)
W12X 79
W12X79tTruss12i
4Tfuss121
Figure 10: Bridge Model with Auto Sections Assigned
20
W12)?9 iTussl2
W1WX791,uss12
?
W1jx9
rus-12,
1o04 Paterns
Load Pattern tlaw
DEAD
CcCk To:
Setf WeIId
Muiu
Type
DEAD
Auto Lateral
Load Patern
AddWW Loed Patern
v i
LOW
PSIM
LPJEI
Show LOadPatern Ns..s
f-OW
Canc
-------- -----
Figure 11: Load Pattern Definition
Cck
Load Coses
Load Case Haea
Load Case Typ
Load COaMOMn
t:
Lo
AN
Add
d
Case
CUM
Add how Cves
LMUSTL2
CUO3Ad
Copy of Load Case
Cofo
Coao
Mompshow co"
ModpyoShw Load Cas.
OiWS Lowd Case
dnoatdmio DftCal
Show Load Case re.
.4.4.
Figure 12: Load Case Definition
CFg"
DEAD
L~ELOWN
T14:
Lvo~WSWC
Suwe
Sic
eO
aomh
ACy
Figure 13: Load Combination Definition
e
R
92 on 1
,
Oft,
$--
",
26
v
21 mu*v
20
%WUKV
&""aUOW Opt*"A~
'W
ay f SM, 4O
Cm
Figure 15: Steel Frame Design Preferences
Figure 14: Linear Static Analysis Definition
21
.K
Analysis and design sections differ for 55 steel frames. Do you want to
select them?
Yes
All steel frames passed the stress/capacity check.
OK
No
Figure 16: Analysis versus Design Section Check
Figure 17: Steel Member Check
2.1.2 Removal of Pier
Once the member sizes for the entire bridge were determined, one of the central supports
was removed from the structure, as shown in section 6.1 of the appendix. This would simulate
the collapse of a pier due to an impact from a vessel. After the support was removed, a linear
static analysis (step 8 from above) was run again. The structure was then checked again to see if
any of the members have failed due to the new demand exceeding the designed capacity, as in
step 10. If any members had failed, the bridge was redesigned using two methods. The first
method designed the structure to be more robust, whereas the second method made use of
smaller, redundant members.
2.1.2.1 Design for Robustness
When designing for robustness, the failed members were increased in size until they were
large enough to support the applied uniform load. Some of the members that did not fail were
also enlarged to ensure that the bridge remained symmetrical along the vertical axis. The
detailed procedure is as follows:
12. Select the failed members and assign them to the Auto Section Property as in step 4.
13. Repeat steps 8 and 10 until all analysis and design sections match and all members pass.
Then select the newly designed members and click Make Auto Select Section Null.
14. Verify that the frame section sizes are identical along the vertical axis of symmetry. For the
newly enlarged members, their symmetrical counterpart should be reassigned to the same section
size.
15. Record the self-weight of the new bridge model, as in step 11.
22
2.1.2.2 Design for Redundancy
When designing the bridge to be redundant, more members were added to the original
bridge design. For the Warren truss bridges, vertical members were added in between each
diagonal member, as seen in the appendix. For the K-truss and the Pratt truss bridges, the span
of each bay was reduced to half of its original size, almost doubling the number of members used
in each bridge. Each bridge was then analyzed in a similar manner as before. After a central
support was removed, the appropriate member sizes were chosen to ensure that the bridge would
not fail under the uniform loading. The detailed procedure is as follows:
16. For the Warren truss models, add vertical elements to each bay. For the K-truss and Pratt
truss models, delete all of the the vertical and diagonal elements. Then add elements so that the
span of the original bay is reduced in half, as shown in section 6.1 of the appendix.
17. Assign all elements to the Auto Section Property, and then follow steps 13, 14, and 15.
2.1.2.3 Comparison of Design Methods
Once the member sizes for each bridge design were chosen, the weights of each structure
were compared to the original bridge weight prior to the support removal. The difference in steel
weight could be equated to the cost of designing the bridge structure to withstand a pier collapse.
The costs associated with robust designs were then compared directly with the costs associated
with redundant designs.
2.1.3 Removal of Members
Aside from a central pier removal, the bridge models were analyzed after removing 1, 2,
and 3 bridge members. This would simulate the removal of members due to vehicle or train
impacts. After one member was removed near the center of the span, the structure was analyzed
for redundancy and robustness as before. Where the removed member meets the bridge deck, a
second member that meets at the same point was also removed. The bridge was then reanalyzed
for the removal of the second member. If a third member also meets at that same point, the
member was removed and the bridge was reanalyzed. The point where all of the removed
members connect could be the collision point of the vehicle or train. As before, the two design
methods were compared after each member removal, based on the amount of steel required for
each design.
23
2.2 Truss Model (Matlab)
The second part of the analysis procedure employs Matlab to run a truss model study on
the bridge models previously used. A Matlab script was written to run the displacement method
on a given truss. The inputs for the program include the geometry of the structure, the crosssectional areas of each member, the loads applied, and the support conditions. The outputs
include the forces in each member, the reactions of each support, and the displacements of each
node.
2.2.1 Point Loads
For each bridge model, the individual members were each given the same cross sectional
area. The uniform factored live and dead load found earlier was converted into point loads by
multiplying the load by the tributary length between bays. The point loads were then applied at
each node along the bridge deck. The Matlab script was run, and the axial forces and reaction
forces were recorded.
2.2.2 Removal of Pier, Full and Half Load
To simulate the removal of a bridge pier, one of the central supports was removed from
the script of each bridge model. The script was run again, and the axial and reaction forces were
recorded. The point loads were then reduced to half of the originally applied loads, to account
for dead load only. This is to account for the fact that in the case event of an extreme event, it is
unlikely that the bridge supports the entire live load it was designed for. When only half of the
load is applied, it is assumed the bridge must only be able to support its self-weight to keep from
undergoing failure.
This method was implemented for both the original bridge design and the redundant
bridge design. Once the pier was removed, the original design was considered to be robust. The
absolute values of the forces in each bridge member was compared for the robust and redundant
bridges to the original bridge prior to the removal of the support.
2.2.3 Removal of 1 Member, Full and Half Load
The bridge models were also analyzed after removing a single bridge member, to
simulate the damage caused by a train or vehicle impact. After one member was removed near
the center of the span, the axial and reaction forces for the structure were recorded. As before,
24
the point loads were reduced to half of the original loads, and reanalyzed. This method was
implemented for both the robust and the redundant bridge designs, for comparison.
2.3 Comparison of Analysis Methods
The truss model method varies from the frame model method because the displacement
method does not account for the same design criteria as SAP2000 employs. For example,
buckling is not accounted for in the truss model method. The SAP2000 models all of the
members as frame elements that are capable of taking bending. The Matlab script models all of
the members as bar elements. Therefore, the uniform loading had to be converted and entered as
point loads.
In the Matlab script, all of the bridge members were assumed to have the same cross
sectional areas. The deflections of the nodes were ignored, and only the forces in each member
and the reaction forces were recorded. Additionally, the forces in the members were compared
directly for each bridge design under the different impact conditions. Previously in the frame
model study, the member sizes which correspond with the forces in the member were increased
evenly on both halves of the structure to ensure a vertical symmetry.
Figure 18: Warren Truss 1.5:1 after running the Design/Check of Structure
Figure 19: Warren Truss 1.5:1 after a support was removed
25
Figure 20: Warren Truss 1.5:1 after resizing of members
3 Results
Analyses have been performed for a variety of 2D truss frames based on varying spans,
topologies, numbers of members, and loading conditions. Each bridge was analyzed for
robustness and redundancy using the analysis procedures previously discussed. These
procedures were used to compare the structural weights of the robust and redundant bridge
designs under parametric conditions. The total weight of steel for each bridge type under the
specified damage and load conditions was compared to the original undamaged bridge weight.
The new weights were converted to a ratio of the original weight. The ratio of each redundant
bridge weight to its respective robust bridge weight was then calculated for each damage
condition. These values are summarized in tables Table 1 through Table 18 below.
3.1 Specifications
The 12 truss bridges shown in figures Figure 21 to Figure 32 were used for the analysis
cases. Each bridge has a vertical depth of 22.4 feet, and the span dimensions are shown for each
bridge below. The width for each bay was 25 feet. The steel was assumed to have a modulus of
elasticity of 29,000 ksi and a yield stress of 50 ksi.
26
100'
00'
so'
100'
75'
100'
250'
275'
Figure 21: Dimensions of 0.5:1 Warren Truss
Figure 22: Dimensions of 0.75:1 Warren Truss
100'
125'
100'
325'
Figure 23: Dimensions of 1:1 Warren Truss
100'
Figure 24: Dimensions of 1.25:1 Warren Truss
150'
100
350'
Figure 25: Dimensions of 1.5:1 Warren Truss
I MY'
175'
10MY
375'
Figure 26: Dimensions of 1.75:1 Warren Truss
I00w
1 0N
I0M'
Figure 28: Dimensions of 1.5:1 K-Truss
Figure 27: Dimensions of 1:1 K-Truss
I00M
I MY
3 50(r1
O
200'
4W0
Figure 29: Dimensions of 2:1 K-Truss
27
Ot'
-
1
I
A'
I,
TN'
I (NT
W
125'
~"
K
100'
7I
3FeT
Figure 31: Dimensions of 1.25:1 Pratt Truss
Figure 30: Dimensions of 1:1 Pratt Truss
100'
150'
100'
350'
Figure 32: Dimensions of 1.5:1 Pratt Truss
Each of the bridge models shown above were used for the robust bridge design.
Examples of the respective redundant bridge designs for each bridge topology are shown in
figure X below. Each model shows the designs for the 1.5:1 span ratio.
77\
/Iyy-
zNJUI=
111\14\=\
Figure 33: Sample Robust and Redundant Designs for All Truss Topologies
3.2 Summary
Tables Table 1, Table 2, and Table 3 show the structural weights that were determined
using SAP2000 of the three types of truss models that were investigated in this thesis. The first
column shows the center-to-side span ratio of the bridge model. The second and third columns
show the structural weights for the robust and the redundant bridge models. The weights shown
were taken from the original, undamaged bridges which were designed for the factored full live
and dead uniform loads. The final column shows the ratio of the weight of the redundant bridge
design to the robust bridge design.
28
It can be seen that for each bridge type, the weights of the bridge increase with the span
ratio. This relationship is portrayed in Figure 34. It can be seen that for each center-to-side span
ratio, each truss type behaves differently. For example, at a span ratio of 1.25:1, the K-truss has
the lightest redundant bridge design but has the heaviest redundant design at a ratio of 2:1. The
robust design for the Pratt truss is lighter than the robust design for the Warren truss for small
span ratios, and then crosses over at around a 1:1 ratio.
For the Warren and Pratt truss bridges, the redundant designs weigh more than the robust
designs for each span. For the K-truss however, the redundant bridge is lighter than the robust
bridge for the 1:1 and 1.5:1 center-to-side span ratios, and then becomes heavier for the 2:1 span
ratio. The redundant-to-robust weight ratio remains around a constant 1.2 for the Warren truss
bridge, and a constant 1.16 for the Pratt truss bridge. The redundant-to-robust weight ratio for
the K-truss increases as the center span increases. These trends could be seen in Figure 35. The
figure suggests that the trends for the Warren and the K-truss cases not perfectly linear, which
could be attributed to the fact that discrete sizes were chosen for the members.
Table 1: Original Weights of Warren Truss Bridge Models in Kips
C:S Span Ratio
Robust
0.5:1
38.6
0.75:1
1:1
1.25:1
1.5:1
1.75:1
42.77
46.81
51.68
58.15
65.99
Redundant Red/Rob
1.20
46.21
1.20
51.35
1.20
55.99
1.21
62.53
1.20
69.57
1.18
78.00
Table 2: Original Weights of K-Truss Bridge Models in Kips
C:S Span Ratio
1:1
1.5:1
2:1
Robust
57.34
70.18
90.21
Redundant
53.98
69.26
92.22
Red/Rob
0.94
0.99
1.02
Table 3: Original Weights of Pratt Truss Bridge Models in Kips
C:S Span Ratio
1:1
1.25:1
1.5:1
Robust
46.62
53.50
61.20
Redundant Red/Rob
1.16
54.09
1.16
62.13
1.16
71.05
29
Bridge Weight vs. C:S Span Ratio
100.80
90.80
80.80
a)
co
70.80
Warren Robust
a
60.80
-
Warren Redundant
ron
E
50.80
-
K Robust
40.80
-
K Redundant
a)
30.80
C|O
Pratt Robust
20.80
Pratt Redundant
10.80
0.80
0.5
0.75
1
1.25
1.5
1.75
2
Center to Side Span Ratio
Figure 34: Weight of Undamaged Bridge vs C:S Span Ratio for All Truss Models
Original Bridge Designs
1.25
o
A
0
1.20
1.15
1.10
C 1.05
-
1.00
0
0
CC
Warren
-PrK
0.95
Pratt
0.90
0.85
0.80
0.5
0.75
1
1.5
1.25
1.75
Center to Side Span Ratio
Figure 35: Weight Ratio of Undamaged Bridge vs. C:S Span Ratio for All Truss Models
30
3.3 Analysis of Warren Truss using Frame Model
Tables Table 4 through Table 9 show the weight ratios of the Warren truss for both the
robust and the redundant bridge designs as compared to its respective undamaged bridge design.
The first column describes the bridge's state of damage. The damage conditions include the
removal of a central support, or the removal of 1, 2, or 3 elements that meet at a single joint. The
second and third columns show the weight ratios for the damaged bridge designs to the original
bridge weight. The final column shows the ratios of the weight of the redundant bridge design to
the robust bridge design.
3.3.1 Warren Truss after Support Removal
The weight ratios of the new bridge designs after damage to the original bridge weight
can be interpreted as the amount of material that would be needed in order to prevent the bridge
from failing. For example, when a pier was removed from the 1:1 Warren truss bridge, the
weight ratio for the robust bridge was 1.7122. This means that in order to resist failure, the
structure would require roughly 70 percent more material than the original design called for. The
1.25:1 Warren truss and the 1.5:1 Warren truss would require about 95 percent and 109 percent
more steel respectively, under the condition that a support is removed. It can be seen that as the
center-to-side span ratio increases, the percentage of steel required to prevent collapse increases.
3.3.2 Warren Truss after Member Removal
For the Warren truss bridge models with 0.5:1, 1:1, and 1.5:1 center-to-side span ratios,
the central joint was chosen as the point of removal for the bridge members. For the bridges
with 0.75:1, 1.25:1, and 1.75:1 center-to-side span ratios, the joint just to the left of center was
chosen as the point of removal for the bridge members. This variance in element removal is
attributable to the topology of the Warren trusses. Some members were removed from the left of
center because there are no elements meeting at the center of the bridge deck for some of the
bridge models. Because of this difference in removal location, the aforementioned bridge spans
will be analyzed separately for the Warren trusses under the member removal conditions.
For the robust bridge design, only two members meet at a joint on the bridge deck,
whereas the redundant design has three members. Therefore, the robust design was only
31
analyzed after the removal of two members. To calculate the redundant to robust weight ratio,
the weight of the redundant bridge after the removal of three members was compared to the
weight of the robust bridge after the removal of two members. This follows the fact that in an
event such as a heavy vehicle collision with the bridge at a certain joint, these are the maximum
amount of members that would be directly damaged by the impact.
Looking at the 0.5:1 Warren truss, the weight ratio for the robust bridge was 1.0470 after
the removal of one member, and 1.2207 after the removal of two members. For the redundant
bridge, the weight ratios were 1.1971, 1.2117, and 1.2245 after the removal of one, two, and
three members respectively. For each design, the bridge requires about 22 percent more steel to
withstand damage to all of the bridge elements leading to one joint.
It is also interesting to note that for the removal of members from the truss bridges, the
weight ratio for the redundant design of the off center member removal (Figure 38) is relatively
even after the removal of one and two members, and then increases after the third member is
removed. For the removal of the central members, the weight ratios for the redundant design is
about the same after all three members are removed (Figure 39).
Table 4: Weight Ratios of 0.5:1 Warren Truss under Parametric Conditions
Conditions
Remove 1
Remove 2
Remove 3
Robust
1.0470
1.2207
N/A
Redundant Red./Rob.
1.1971
1.1434
1.2117
0.9926
1.2245
1.0031
Table 5: Weight Ratios of 0.75:1 Warren Truss under Parametric Conditions
Conditions
Remove 1
Remove 2
Remove 3
Robust
1.0000
1.2879
N/A
Redundant Red./Rob.
1.2207
1.2207
1.2207
0.9479
1.2709
0.9868
Table 6: Weight Ratios of 1:1 Warren Truss under Parametric Conditions
Conditions
Remove Support
Remove 1
Remove 2
Remove 3
Robust
1.7122
1.0786
1.2182
N/A
Redundant Red./Rob.
1.7748
1.0366
1.2177
1.1290
1.2177
0.9996
1.2177
0.9996
32
Table 7: Weight Ratios of 1.25:1 Warren Truss under Parametric Conditions
Conditions
Remove Support
Remove 1
Remove 2
Robust
1.9511
1.0000
1.3211
Redundant
2.0376
1.2138
1.2138
Red./Rob.
1.0443
1.2138
0.9188
Remove 3
N/A
1.2858
0.9733
Table 8: Weight Ratios of 1.5:1 Warren Truss under Parametric Conditions
Conditions
Robust
Remove Support 2.0892
Remove 1
1.1026
Redundant
2.4186
1.1889
Red./Rob.
1.1577
1.0783
Remove 2
1.2179
1.1889
0.9762
Remove 3
N/A
1.1919
0.9787
Table 9: Weight Ratios of 1.75:1 Warren Truss under Parametric Conditions
Conditions
Robust
Redundant
Red./Rob.
Remove 1
1.0000
1.1873
1.1873
Remove 2
Remove 3
1.2960
N/A
1.1937
1.2881
0.9211
0.9940
Warren Truss after Removal of Members (SAP2000)
1.3
C
0
4_1.
_0
.~1.1
--
_0
---
~009
W 0.5:1
W 1:1
1.5:1
Original Design
4-
-r-0.8
0
1
3
2
Number of Elements Removed
Figure 36: Weight Ratio of Bridge vs. Number of Elements Removed for Warren Truss Models
33
Warren 2 Truss after Removal of Members (SAP2000)
1.3
a)
ao 1.2
co 1.1
0
-
W2 0.75:1
-
W2 1.25:1
0
1.0
W2 1.75:1
-
Original Design
0.9
',
0
4-
3
2
0
Number of Elements Removed
Figure 37: Weight Ratio of Bridge vs. Number of Elements Removed for Warren 2 Truss Models
Warren Truss after Removal of Members (SAP2000)
1.30
1.25
0
0
1.20
W 0.5:1 Robust
a)
E 1.15
0
1.10
W 0.5:1 Redundant
--
W 1:1 Robust
-
W 1:1 Redundant
--
W 1.5:1 Robust
-
W 1.5:1 Redundant
/
-
-
0
1.05
1.00
0
1
2
3
Number of Elements Removed
Figure 38: Weight Ratio of New Bridge to Original Bridge vs. Number of Elements Removed for
Warren Truss Models
34
Warren 2 Truss after Removal of Members (FEA)
1.35
S 1.30
0 1.257
W2 0.75:1 Robust
.-
1.2
-W2
0.75:1 Redundant
E 1.15
U
-
W2 1.25:1 Robust
4.1
-
W2 1.25:1 Redundant
W2 1.75:1 Robust
1.05
.0FU 1.00
W2 1.75:1 Redundant
0.95
0
1
3
2
Number of Elements Removed
Figure 39: Weight Ratio of New Bridge to Original Bridge vs. Number of Elements Removed for
Warren2 Truss Models
3.4 Analysis of K-Truss using Frame Model
Tables Table 10 through Table 12 show the weight ratios of the K-truss for both the
robust and the redundant bridge designs as compared to its respective undamaged bridge design.
Again, the first column describes the bridge's state of damage. The second and third columns
show the weight ratios for the damaged bridge designs to the original bridge weight, and the final
column shows the ratios of the weight of the redundant bridge design to the robust bridge design.
3.4.1 K-Truss after Support Removal
When a pier was removed from the 1:1 K-truss bridge, the weight ratio for the robust
bridge was 1.8368. This means that in order to resist failure, the structure would require roughly
84 percent more material than the original design called for. The 1:1 redundant design requires
less steel, at 75 percent more. However, for the 2:1 K-truss, the robust design requires less
material than the redundant design, at 227 percent and 250 percent respectively. This helps
demonstrate the relationship that as the center-to-side span ratio increases, the robust design
becomes lighter than the redundant design.
35
3.4.2 K-Truss after Member Removal
For each of the K-truss bridge models, the central joint was chosen as the point of
removal for the bridge members. Unlike with the Warren truss, both the robust and the
redundant bridge designs have three members that meet at a single joint on the bridge deck.
Therefore, the two bridge designs could be directly compared after the removal of one, two and
three members.
Looking at the K-trusses after each individual member removal, it can be seen that the
weight ratios for the robust design increase as the span increases. The weight ratios of the
redundant design also increase with the span, but at a faster rate. However, the redundant bridge
weight does not change significantly as the number of removed members increases, whereas the
robust weight increases. The redundant to robust weight ratio is below 1 for all of the K-truss
bridges examined, meaning that the redundant design is lighter than the robust design for each
member removal. However, the undamaged redundant bridge design for the 2:1 span is heavier
than the robust design, which was presented in Table 2.
Table 10: Weight Ratios of 1:1 K-Truss under Parametric Conditions
Conditions
Robust
Remove Support 1.8368
Remove 1
1.0360
Remove 2
1.0945
Remove 3
1.1555
Redundant Red./Rob.
1.7501
0.9528
0.9448
0.9119
0.9526
0.8704
0.9568
0.8280
Table 11: Weight Ratios of 1.5:1 K-Truss under Parametric Conditions
Conditions
Remove Support
Remove 1
Remove 2
Remove 3
Robust
2.2724
1.0553
1.1200
1.1299
Redundant Red./Rob.
2.4971
1.0989
0.9996
0.9473
1.0047
0.8971
1.0118
0.8955
Table 12: Weight Ratios of 2:1 K-Truss under Parametric Conditions
Conditions
Remove 1
Remove 2
Remove 3
Robust
1.0531
1.1395
1.1529
Redundant Red./Rob.
1.0273
0.9754
1.0325
0.9061
1.0325
0.8956
36
Removal of Members (K-Truss)
1.1
ca
1.0
0
-K
1:1
C
an
0
0.9
0
-
K 1.5:1
-K
2:1
-Original
Design
0-
0.8
0
3
1
Number of Elements Removed
Figure 40: Weight Ratio of Bridge vs. Number of Elements Removed for K-Truss Models
Removal of Members (FEA)
1.20
.C:
.9.
1.15
K 1:1 Robust
1.10
-K
0.
E
1.05
1:1 Redundant
-
U
-
1.00
K 1.5:1 Robust
K 1.5:1 Redundant
K 2:1Robust
0
o 0.95
-
K 2:1 Redundant
0.90
0
1
3
2
Number of Elements Removed
Figure 41: Weight Ratio of New Bridge to Original Bridge vs. Number of Elements Removed for KTruss Models
37
3.5 Analysis of Pratt Truss using Frame Model
Tables Table 13 through Table 15 show the weight ratios of the Pratt-truss for both the
robust and the redundant bridge designs as compared to its respective undamaged bridge design.
Again, the first column describes the bridge's state of damage. The second and third columns
show the weight ratios for the damaged bridge designs to the original bridge weight, and the final
column shows the ratios of the weight of the redundant bridge design to the robust bridge design.
3.5.1 Pratt-Truss after Member Removal
For each of the Pratt-truss bridge models, the central joint was chosen as the point of
removal for the bridge members. Like with the K-truss, both the robust and the redundant bridge
designs have three members that meet at a single joint on the bridge deck. Again, the two bridge
designs could be directly compared after the removal of one, two and three members.
Looking at the Pratt trusses after the removal of each member, it can be seen that the
weight ratios for the robust design increase as the number of removed members increases. The
weight ratios of the redundant design also increase, but at a slower rate. The redundant to robust
weight ratio is generally above 1 for the removal of one or two members, and below one for the
removal of two or three members. This means that the redundant design becomes lighter than
the robust design as more members are removed.
Table 13: Weight Ratios of 1:1 Pratt Truss under Parametric Conditions
Conditions
Robust Redundant Red./Rob.
Remove 1
Remove 2
Remove 3
1.0978
1.1840
1.2489
1.1715
1.1784
1.1860
1.0672
0.9953
0.9496
Table 14: Weight Ratios of 1.25:1 Pratt Truss under Parametric Conditions
Conditions
Remove 1
Remove 2
Remove 3
Robust
1.0783
1.1324
1.1949
Redundant Red./Rob.
1.1687
1.0838
1.1727
1.0356
1.1774
0.9853
38
Table 15: Weight Ratios of 1.5:1 Pratt Truss under Parametric Conditions
Conditions
Remove 1
Remove 2
Robust
1.0774
1.1346
Remove 3
1.1838
Redundant Red./Rob.
1.0875
1.1717
1.0387
1.1786
1.1826
0.9990
Removal of Members (Pratt)
a)
1.2
0x
1.1
0
0
-
1.0
0
0-
P 1:1
P 1.25:1
0.9
-
P 1.5:1
-
Original Design
0.8
0
1
2
3
Number of Elements Removed
Figure 42: Weight Ratio of Bridge vs. Number of Elements Removed for Pratt Truss Models
39
Removal of Members (FEA)
1.25
C
1.20
P 1:1 Robust
1.15
P 1:1 Redundant
E
0
-FD
P 1.25:1 Robust
1.10
-
P 1.25:1 Redundant
1.5:1 Robust
0~P
1.05-P
1.5:1 Redundant
1.00
0
1
3
2
Number of Elements Removed
Figure 43: Weight Ratio of New Bridge to Original Bridge vs. Number of Elements Removed for
Pratt Truss Models
3.6 Analysis of Warren Truss using Truss Model
Tables Table 16 through Table 18 show the axial force ratios of the Warren truss for both
the robust and the redundant bridge designs as compared to its respective undamaged bridge
design. These data were determined using the displacement method that was encoded into
Matlab. The force ratios were computed by summing the absolute values of the axial forces for
all of the members before and after damage. Then the force total of the damaged state was
divided by the force total of the pre-damaged state. The first column describes the bridge's state
of damage, under both the full load and a reduced load. The second and third columns represent
the ratios of the total combined axial forces for the damaged bridge designs to the forces of the
undamaged bridge. The final column shows the ratios of the forces of the redundant bridge
design to the robust bridge design.
3.6.1 Comparison to Frame Model
As compared to the weight ratios of the Warren trusses found using the frame model
analysis method, the force ratios found by the displacement method are higher for each
respective condition. In particular, the force ratios for the redundant bridge designs are much
40
higher than those of the robust designs. These differences can be attributed to the fact that the
two analyses are based on different assumptions, as highlighted in section 2.3.
Despite the differences between the two analysis procedures, the forces found by the
displacement method follow a similar trend to the bridge weights found earlier. For example,
when a central support is removed, the force ratio of the redundant design increases faster than
that of the robust design with the span ratio. The Warren truss follows the same relationship
after the removal of a member.
3.6.2 Warren Truss under Reduced Load
The forces in the bridge models were also analyzed under half of the initially applied
load. This would simulate the bridge under dead load only, as the factored live load is roughly
half of the total factored load that was applied to the bridge models. The reason for analyzing
this is that in the case of an extreme event, the live load may not be at its maximum. Therefore it
is important to establish a relationship between the forces in the structure after damage and the
amount of live load that is applied at the time.
For each bridge span under each condition, the force ratio under half of the load was
roughly half of the ratio found under the full load. This suggests that the forces in the bridge
members are directly related to the applied load, meaning that the force ratio varies linearly with
the loading applied.
Table 16: Force Ratios of 1:1 Warren Truss under Full and Reduced Loads
Conditions
Robust Redundant Red./Rob.
1.6723
4.5133
Remove Support 2.6989
Half Load
Remove 1
Half Load
1.3440
1.0876
0.5418
2.1984
1.6734
0.8024
1.6357
1.5387
1.4810
Table 17: Force Ratios of 1.25:1 Warren Truss under Full and Reduced Loads
Conditions
Remove Support
Half Load
Remove 1
Half Load
Robust
3.0856
1.5350
1.1000
0.5500
Redundant
5.3974
2.5909
1.7000
0.8500
Red./Rob.
1.7492
1.6879
1.4286
1.4277
41
Table 18: Force Ratios of 1.5:1 Warren Truss under Full and Reduced Loads
Redundant Red./Rob.
1.8083
6.2881
Conditions
Remove Support
Robust
3.4774
HalfLoad
Remove 1
1.6261
1.1351
2.8303
2.0526
1.7405
1.8082
Half Load
0.5584
0.9071
1.6245
3.7 Cost Analysis
3.7.1 Cost of Withstanding Removal of Pier
For the purpose of this thesis, the cost of material is assumed to be directly related to the
weight of steel required for each bridge. When looking strictly at the bridges designed to
withstand the removal of a pier, the percentage of additional steel required was as low as 71
percent for the robust 1:1 Warren truss bridge. The maximum percentage of material that was
obtained was 150 percent for the redundant 1.5:1 K-truss bridge. For both bridge types, the cost
of additional steel required varies linearly with the center-to side span ratio. This can be seen in
Figure 44.
Removal of Support (SAP2000)
2.6
2.4
2.2
0
2.0
~0
1.8
CD
1.6
0.
1.4
4E
Warren Robust
-
Warren Redundant
K-Truss Robust
-
K-Truss Redundant
1.2
-
1.0
Original Design
0.8
1
1.1
1.2
1.3
1.4
1.5
Center to Side Span Ratio
Figure 44: Weight Ratio of New Bridge to Original Bridge vs. C:S Span Ratio after the Removal of
a Support
42
3.7.2 Cost of Withstanding Removal of One Member
When looking at the bridges designed to withstand the removal of a single member, the
cost of additional steel required was as low as 5 percent for the robust 0.5:1 Warren truss bridge.
The redundant 0.5:1 Warren truss requires about 20 percent more steel. However, as the span
increases, the robust Warren bridge designs require a higher percentage of steel, whereas the
redundant designs require a smaller percentage. For the Pratt bridge type, the cost of steel
required to support the damage of one member remains relatively constant with the span ratio for
both bridge designs. In general, the robust Pratt design costs less than the redundant design
under this condition. For the K-truss, the redundant bridge design costs less than the robust
bridge design for all of the spans that were studied. However, as the span ratio increases, the
redundant bridge requires a higher material percentage while the robust bridge percentage
remains relatively constant. Each of these relationships can be seen in Figure 45.
Removal of One Member (SAP2000)
_
1.25
.: 1.20
o 1.15
Warren Robust
0
.S
--
S1.05
__
.--
Warren Redundant
K Robust
E 1.00
U
-:
-
0.95
Pratt Robust
0.90
0
-C
K Redundant
-
0.85
-Original
o.8W
0.5
Pratt Redundant
0.75
1
1.25
1.5
1.75
Design
2
Center to Side Span Ratio
Figure 45: Weight Ratio of New Bridge to Original Bridge vs. C:S Span Ratio after the Removal of
One Member
3.7.3 Cost of Withstanding Removal of Two Members
When looking at the bridges designed to withstand the removal of two members, the cost
of additional steel required for the robust Warren truss remains constant across the span ratios.
This differs from the removal of one member, where the cost percentage of material increased
43
with the span ratio. As before, the required material for the redundant design decreases with the
span ratio. For the K-truss and the Pratt truss, the relationships between cost percentage of steel
and the span ratio is similar to that of a single member removal for both bridge designs.
However, while the gap between the cost ratios of the K-truss bridge type opens after the
removal of the second member, the gap between the Pratt bridge type closes. These relationships
are shown in Figure 46.
Removal of Two Members (SAP2000)
1.30
1.25
0 1.20
.6
0
1.15
-
Warren Robust
_0
a
-
Warren Redundant
E
0
1.10
K Robust
U
1.05
V)
4-
1.00
-
Pratt Robust
4-0
a)
K Redundant
0.95
0.90
-
Pratt Redundant
-
Original Design
0.85
0.5
0.75
1
1.25
1.5
1.75
2
Center to Side Span Ratio
Figure 46: Weight Ratio of New Bridge to Original Bridge vs. C:S Span Ratio after the Removal of
Two Members
3.7.4 Cost of Withstanding Removal of Three Members
The relationships between the steel cost ratio and the center-to-side span ratio for each
bridge after the removal of three members remains relatively similar to those after the removal of
two members. The most notable difference is that the robust Pratt bridge becomes more
expensive when a third member is removed, whereas the other bridge types remain around the
same cost. This can be seen in Figure 47.
44
'__- I
............
.
.......
- - - .... .....
Removal of Three Members (SAP2000)
1.30
1.25
1.20
0
-
1.15
E
0
U
-Warren
1.10
1.05
-
Redundant
-
K Robust
-
K Redundant
1.00
0
Warren Robust
-
Pratt Robust
0.95
-
Pratt Redundant
0.90
-
Original Design
0.85
0.5
0.75
1
1.25
1.5
1.75
2
Center to Side Span Ratio
Figure 47: Weight Ratio of New Bridge to Original Bridge vs. C:S Span Ratio after the Removal of
Three Members
3.7.5 Discussion on Truss Model
Figures Figure 48 and Figure 49 portray the percentages of additional steel required for
the Warren truss bridges after undergoing damage that were found using the displacement
method. As compared to the results found using the frame model analysis method, the
percentages are higher. However, the trends for the percentage of steel versus the span ratios are
comparable. Since the assumptions made for the frame models are closer aligned with the actual
behavior of bridges, the structural weight ratios found by the preceding method will be assumed
as the acceptable values.
45
..........
.....
....
...
..
..
Removal of Support (Matlab)
_ 6.8
.C:
0 5.8
0
0
4.8
E
-Warren
-. 3.8
0
U
Robust
-
Warren Redundant
-
Warren Robust Half
-
Warren Redundant Half
-
Original Design
2.8
E 1.8
0.8
1
0
0L
1.1
1.2
1.3
1.4
1.5
Center to Side Span Ratio
Figure 48: Force Ratio vs. C:S Span Ratio for Warren Truss Models after the Removal of a
Support
Removal of Member (Matlab)
2.3
2.1
_0
1.7
-
Warren Robust
1.3
-
Warren Redundant
1.1
-
Warren Robust Half
.
0
1.9
1.5
E
0
U
0.9
-Warren
E 0.7
0 0.5
-
Redundant Half
Original Design
0.3
1
U
1.1
1.2
1.3
1.4
1.5
Center to Side Span Ratio
Figure 49: Force Ratio vs. C:S Span Ratio for Warren Truss Models after the Removal of a
Member
46
3.7.6 Additional Notes on Cost
As mentioned earlier, the weight of the required structural steel for each bridge has been
directly correlated to the cost of material for the bridge. However, it is important to note that the
amount of steel is not the only factor that is included in the cost of the bridge. The number of
members and connections, the regularity of connections and the simplicity of design can all
contribute to the cost of the bridge. For example, while the redundant bridge designs may
require less steel than the robust designs in certain cases, they call for more members and
connections. More connections mean that more work has to be done to construct the bridge. In
areas where labor is inexpensive relative to the cost of material, it may be cheaper to go with the
redundant design. However, in areas where labor costs are high and material costs are relatively
low, the robust design may be the less expensive route.
Other costs that are associated with constructing a bridge include land acquisition,
planning, architectural and engineering design, equipment, labor, materials, insurance and taxes,
inspections, field supervision, and overhead costs (Hendrickson & Au, 1989).
Most of the non-
material costs are not affected by the resiliency aspects that have been previously discussed.
Additionally, the cost of the superstructure is only a portion of the total material cost. Based on
11 different itemized bridge costs from Infrastructure Project Estimating, the superstructure costs
have ranged from about 10 percent to 70 percent of the overall costs of materials (Item Cost
Summary).
3.8 Probability of Collapse
According to AASHTO design procedures, it is required for bridge engineers to assess
the risk of structural failure and collapse associated with each bridge design. This risk
assessment must be made on the basis of the bridge's environmental conditions, i.e. waterway
traffic, climate, unforeseen loads it may be subjected to, etc. The risk assessment of a bridge
collapse is expressed as the percent chance of occurrence over a one-year period, due to a given
factor. The inverse of this number yields the predicted number of years that will pass before an
occurrence will induce collapse, otherwise known as the return period. While certain factors
may occur more often than the return period suggests, the bridge will be able to withstand such
47
loads for a certain amount of time. For factors that are independent of one another, i.e.
earthquakes and boat impacts, and because collapse probabilities are always small in practice,
the total probability of collapse during a one-year period is the sum of the probabilities of
collapse for each individual factor.
3.8.1 Vessel Collision
For bridges spanning navigable waterways, it is necessary to account for vessel collisions
in the design. Each time a boat passes below a bridge, there is an associated risk the boat will
impact a structural component of the bridge. Factors that affect the likelihood of such risks
include boat traffic through the waterway, the span of the bridge in between piers, the size of the
boats, and the speed of the boats. During a collision, the boat imparts a dynamic load to the
bridge, most likely a support. If the impact is large enough, the collision can lead to the collapse
of one of the supports. In turn, this can lead to the progressive collapse of the entire structure.
The AASHTO guide book provides a section on designing bridges to withstand vessel
collisions. According to the code, bridge structures should either provide protection against
collision forces by the implementation of fenders, dikes, or dolphins, or should be designed to
withstand such a collision. The minimum design impact load should be determined using an
empty hopper barge that is moving at a velocity equal to the mean yearly current for the
waterway location. The barge should have a dimension of 35ft x 195ft, with a weight of 200
tons.
The AASHTO provisions contain a section on the barge force-deformation relationship
from experimental data that was acquired in the 1980s. Several scaled barges were constructed
and crushed, to develop a relationship between the impact force of a vessel and its respective
bow deformation, as shown in Figure 50 below. PB is the equivalent static barge impact force,
aB is the barge bow damage length, EB is the deformation energy, and PB is the average
equivalent static barge impact force resulting from the study. This data is currently used in
bridge design for barge collision loading.
48
21000
-18000
2500-
-
PB
2000-20
-
29000
.
-1500-
00U
B
CL1000-
500
30
0-0
0
2
4
8
6
10
112
aB (FEET)
Figure 50: Barge Impact Force, Deformation Energy, and Damage Length Data (AASHTO 2012)
For a given barge, the AASHTO provisions could be used to determine the impact load
that an at-risk bridge element should be designed for. Using the weight of a barge (W), its
impact velocity (V), and the hydrodynamic mass coefficient (CH), the kinetic energy (KE) of the
barge could be computed as follows:
CHWV 2
29.2
The weight should be computed in metric tonnes, and the velocity in ft/sec. For each vessel type,
the draft associated with the weight of the vessel is related to its respective hydrodynamic mass
coefficient. The mass coefficient accounts for the mass of the water that is surrounding and
moving with the vessel. CH is taken as 1.05 if underkeel clearance exceeds 0.5 x draft, and 1.25
if underkeel clearance is less than 0.1 x draft. Any values that fall in between can be linearly
interpolated.
The vessel's velocity is dependent not only on its weight and draft, but also its orientation
and distance from its intended transit path. Data regarding vessel traffic data have been taken
from a study of the New St. George Island Bridge conducted at the University of Florida
(Davidson, 2010). The data is presented in Table 19.
49
Table 19: Annually Averaged Vessel Traffic Data for New St. George Island Bridge
Vessel
Group
1
2
3
4
5
6
7
8
N
85
25
117
92
135
22
19
28
Draft
(ft)
2.1
5.6
8.3
11
2
4.9
8.2
11.8
No.
Barges
1
1
1
1
1.9
1.9
1.9
1.9
Bbarge
(ft)
51
58.6
50.6
54
50.9
62.4
45.2
72.4
Lbarge
(ft)
216
316
246
319
267
328
251
256
LOA
(ft)
291
391
321
4399
582.3
698.2
551.9
606.4
W
(tonnes)
971
3288
3259
5907
1777
6026
5945
12346
V
(knots)
5.6
4.6
4.6
4.6
6.4
5.4
5.4
5.4
Using the calculated kinetic energy, the magnitude of the barge deformation is determined by the
following:
aB
10.2
+ KE
5672
RB
RB is equal to LB, where BB is the barge width in feet. A static impact load(PBs) is then
calculated using the following conditions:
PBS =4112aBRB
B= 1(39+1
B)RB
if
aB < 0.34
if
aB
0.34
The probability of collapse (PC) of a bridge pier due to a vessel collision is dependent on
the ultimate lateral resistance of the pier (Hp), the resistance of the superstructure (Hs), and the
impact force of the vessel (P). The AASHTO expressions for calculating PC stemmed from
studies on the damage rates of ship-ship collisions. The ship damage rate could be interpreted as
the ratio of the actual force that is imparted during a collision versus the maximum possible force
that could be imparted. Using the ship-ship collision data, the AASHTO code applies the
collision force ratio to the smaller ship, and assumes the larger ship to represent the bridge pier.
Based on this assumption, a correlation was made between the rate of ship damage and the
capacity-to-demand ratio (C/D) of the bridge's resistance to the impact.
50
The probability of collapse expression that has been developed is the following:
If 0.0 < H/P < 0.1, then
PC = 0.1+ 9 (0.1-P)
If 0.1 < H/P < 1.0, then
PC = 0.111 (1-P)
If H/P;> 1.0, then
PC = 0.0
The probability of collapse distribution curve is shown in Figure 51.
0.
1.0
0.5
Hs or HP
ULTIMATE BRIDGE ELEMENT STRENGTH
VESSEL IMPA0CT FORCE
P,ta, or NP)
Figure 51: Probability of Collapse Distribution (AASHTO 2012)
Using the AASHTO equations, the probability of the collapse of a bridge pier could be
determined using the traffic data from the New St. George Island Bridge. Assuming the bridge
pier has a pushover capacity of 2300 kips, the probability of pier collapse due to each of the 8
vessel groups is shown in Table 20 below.
Table 20: Probabilty of Collapse of Bridge Pier due to Vessel Collision based on Traffic Data for
New St. George Island Bridge
PC
PB (kips)
aB (ft)
KE (kip-ft)
Wimpact
CH
Vessel Group
0
367
0.06
100
1020
1.05
1
0
1419
0.21
390
3452
1.05
2
0
1219
0.21
335
3422
1.05
3
0
2206
0.74
1335
6202
1.05
4
0
2107
0.91
1563
1866
1.05
5
0.022
2878
2.41
5784
6327
1.05
6
0
2050
2.17
3537
6242
1.05
7
0.038
3483
3.04
9156
12963
1.05
8
51
3.8.2 Vehicle Collision
AASHTO also provides a section on designing bridges for vehicle collision. The annual
frequency for a bridge pier to be hit by a heavy vehicle (AFHPB) can be calculated by:
AFHPB =
2(ADTT)
(PHPB) *
365
where ADTT is equal to the number of trucks that pass beneath the bridge per day in one
direction and PHBP is equal to the annual probability for a bridge pier to be hit by a heavy
vehicle. For roadways that are undivided in both tangent and horizontally curved sections, a
3.457 x 10-9 is used. For divided roadways in tangent sections, PHBP is 1-090
PHBP of
x
10-9,
and for divided roadways in horizontally curved sections, PHBP is 2.184 x 10-9. In this thesis,
divided roadways in tangent sections have been explored.
Data on the annual frequency for a bridge pier to be hit by a heavy vehicle is based on
statistical findings by the Texas Transportation Institute. The following table provides some
typical AFHBP values. The last column is of particular interest in this study.
Table 21: Typical Values of the Annual Frequency for a Bridge Pier to be Hit by a Heavy Vehicle
(AASHTO 2012)
ADT Both
Directions
Undivided
Divided Curved
ADTT*
PHBp=3.457E-09 I PHBP=2 .1 8 4 E-0 9
One WayL
150
200
300
400
600
700
800
900
1000
1100
1200
1300
0.0001
0.0003
0.0004
0.0005
0.0008
0.0010
0.0015
0.0018
0.0020
0.0023
0.0025
0.0028
0.0030
0.0033
2 x ADTT x 365
0.0001
0.0002
0.0002
0.0003
0.0005
0.0006
0.0010
0.0011
0.0013
0.0014
0.0016
0.0018
0.0019
0.0021
1400
0.0035
0.0022
AFHPB =
1000
2000
50
100
3000
4000
6000
8000
12000
14000
16000
18000
20000
22000
24000
26000
28000
52
Divided Tangent
PHBP=1.09E-09
x PHBP
0.0000
0.0001
0.0001
0.0002
0.0002
0.0003
0.0005
0.0006
0.0006
0.0007
0.0008
0.0009
0.0010
0.0010
0.0011
4 Conclusions
4.1 Discussion
The two different analysis procedures employed in this study to evaluate resiliency of
truss bridges each have advantages and disadvantages. The frame model analysis procedure is
more precise in that it takes into account realistic steel members that could be used for the actual
design. It accounts for bending, stiffness, and buckling, aside from the stress capacity of each
element. The truss model method, while less accurate, is a quicker and simpler analysis method.
Although the results are not heavily relied upon in this study, the process allows for a quick
assessment of the results from the previous procedure.
The bridge models observed in this study were evaluated before and after they were
subjected to damage. After being damaged, the bridges were redesigned for both robustness and
redundancy. The structural weights of each new bridge design was recorded and compared to
the weight of the original bridge design. For the bridges that were subjected to the removal of a
central support, the percentage of additional material required for the bridge to resist failure
increased linearly with the span of the bridge. For each of the truss topologies that were studied,
the bridge designs with fewer, more robust members required less steel than the designs with
more redundancy. For the spans that were observed, the additional amount of steel required for
the robust design was about 100 percent on average. This means that in order for the bridge to
withstand the removal of a pier, its structural weight would need to double.
The bridge models were also assessed after the damage one, two, and three elements that
meet at a single joint on the bridge deck. For all three truss topologies, the weight ratios of the
redundant bridge design to the robust bridge design decreased as the number of damaged
elements increased. This suggests that as more elements are removed, redundancy becomes
more favorable to robustness. As more members are removed from the redundant designs, the
required material does not change dramatically. For the robust design, the number of elements
removed has a greater effect on the amount of material required. For the observed truss models,
the average required material to withstand damage to a single element is about 7 percent. For the
removal of two and three elements, about 15 and 17 percent is required, respectively.
53
Based on the cases analyzed, it has been evidenced that the cost of designing a truss
bridge for resilience is low relative to the total bridge cost. For each truss topology, there is a
crossover point where the redundant bridge becomes more cost efficient than the robust bridge,
and vice versa. This is contingent upon the bridge span and the number of damaged elements.
4.2 Future Work/Impact
Continued research could help to further develop the relationships between redundancy
and robustness in bridge design. The scope of this thesis was limited to truss bridges of three
different topologies. More truss topologies could be explored, as well as different bridge types.
The depths of the trusses could also be varied, to evaluate how redundancy and robustness vary
with depth. It would also be valuable to analyze three dimensional bridges, because the forces in
the structure would behave differently than those of the two dimensional bridges in the damaged
state. The relationships established in this thesis can be used as a guide to efficiently design
resilient bridges against particular events.
54
5 References
AASHTO LRFD Bridge Design Specifications, Customary U.S. Units. (2012). Washington DC:
American Association of State Highway and Transportation Officials.
Assis, G. (2006, January 23). Impact-ResistantBridges Are Feasible. Retrieved from ENR:
http://enr.construction.com/opinions/viewPoint/archives/060123.asp
Bridge collapses on E18 highway. (n.d.). Retrieved from Norway's News in English:
http://www.newsinenglish.no/2015/02/02/bridge-collapses-on-el 8-highway/
Davidson, M. T. (2010). ProbabilityAssessment of Bridge Collapse under Barge Collision.
University of Florida.
Hancock. (2002, May 26). Towboat Captain Was Short On Sleep. Retrieved from CBS News:
http://www.cbsnews.com/news/towboat-captain-was-short-on-sleep/
Hendrickson, C. T., & Au, T. (1989). Cost Estimation. In ProjectManagementforConstruction.
Englewood Cliffs, NJ: Prentice Hall. Retrieved from Project Management for
Construction.
Hopkins, C. (2002, July 30). Trafficflows again on 1-40 bridge. Retrieved from NewsOK:
http://newsok.com/article/894007
(n.d.). Item Cost Summary Bridge & Overpass Projects. Infrastructure Project Estimating.
Miyachia, K., Nakamurab, S., & Mandaa, A. (2012). Progressive collapse analysis of steel truss
bridges and evaluation of ductility. Journalof ConstructionalSteel Research.
Norwegian bridge collapsepromptsfurther investigations. (2015, February 13). Retrieved from
World Highways: http://www.worldhighways.com/categories/road-highwaystructures/news/norwegian-bridge-collapse-prompts-further-investigations/
/
Welch, W. M. (2013, May 24). No deaths after Wash. state bridge collapses. Retrieved from
USA Today: http://www.usatoday.com/story/news/nation/2013/05/23/washington-skagitriver-bridge-collapse/23 56801
55
6 Appendix
6.1 Robust and Redundant Bridge Models after Damage
Figure 52: Robust and Redundant 0.5:1 Warren Truss after Support Removal
/VVVV\
Figure 53: Robust and Redundant 0.5:1 Warren Truss after Member Removal
Figure 54: Robust and Redundant 0.75:1 Warren Truss after Support Removal
NNNN7
Figure 55: Robust and Redundant 0.75:1 Warren Truss after Member Removal
Figure 56: Robust and Redundant 1:1 Warren Truss after Support Removal
56
Figure 57: Robust and Redundant 1:1 Warren Truss after Member Removal
Figure 58: Robust and Redundant 1.25:1 Warren Truss after Support Removal
Figure 59: Robust and Redundant 1.25:1 Warren Truss after Member Removal
Figure 60: Robust and Redundant 1.5:1 Warren Truss after Support Removal
/N/N/N/N/Nz
//NN//
Figure 61: Robust and Redundant 1.5:1 Warren Truss after Member Removal
Figure 62: Robust and Redundant 1.75:1 Warren Truss after Support Removal
57
/l/lkN
/NN/=
A
/\//\\/
/N/N/N/N/N/
Figure 63: Robust and Redundant 1.75:1 Warren Truss after Member Removal
-A-T
Figure 64: Robust and Redundant 1:1 K-Truss after Support Removal
Figure 65 Robust and Redundant 1:1 K-Truss after Member Removal
Figure 66: Robust and Redundant 1.5:1 K-Truss after Support Removal
Figure 67: Robust and Redundant 1.5:1 K-Truss after Member Removal
Figure 68: Robust and Redundant 2:1 K-Truss after Support Removal
58
IA YNN NNN NNr'~
P'~NN IN.
Figure 69: Robust and Redundant 2:1 K-Truss after Member Removal
T
-
-
N
Figure 70: Robust and Redundant 1:1 Pratt Truss after Support Removal
/ITTW\/M ELLA=
/TNTVP\//IT M/NA=
1TT]\-7VVM7T-VVVNTV7\
Figure 71: Robust and Redundant 1:1 Pratt Truss after Member Removal
Figure 72: Robust and Redundant 1.25:1 Pratt Truss after Support Removal
Figure 73: Robust and Redundant 1.25:1 Pratt Truss after Member Removal
Figure 74: Robust and Redundant 1.5:1 Pratt Truss after Support Removal
59
Figure 75: Robust and Redundant 1.5:1 Pratt Truss after Member Removal
60