By MICHAEL THOMAS DAVIDSON - University of Floridaufdcimages.uflib.ufl.edu/UF/E0/02/14/72/00001/davidson_m.pdf · SIMPLIFIED DYNAMIC BARGE COLLISION ANALYSIS FOR BRIDGE PIER DESIGN

SIMPLIFIED DYNAMIC BARGE COLLISION ANALYSIS FOR BRIDGE PIER DESIGN

By

MICHAEL THOMAS DAVIDSON

A THESIS PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE

UNIVERSITY OF FLORIDA

2007

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© 2007 Michael Thomas Davidson

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To my wife, Kiristen

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ACKNOWLEDGMENTS

This material is based on work supported under a National Science Foundation Graduate

Research Fellowship. However, this thesis would not have been completed without the support

of several individuals. First, the insight and guidance of Dr. Gary Consolazio has proven

invaluable. His willingness to invest time in helping graduate students become effective analysts

and independent researchers will undoubtedly garner countless and vast returns. The author also

wishes to thank Dr. Marc Hoit, Dr. Petros Christou, and Dr. Jae Chung for their assistance with

extending the capabilities of FB-MultiPier. A graduate student deserving of many thanks and

much future success is David Cowan, whose brilliance seems to be limitless. Finally, the author

wishes to thank his wife Kiristen, his family, and his friends for their enduring love and

fellowship.

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TABLE OF CONTENTS

ACKNOWLEDGMENTS ...............................................................................................................4

LIST OF TABLES...........................................................................................................................7

LIST OF FIGURES .........................................................................................................................8

ABSTRACT...................................................................................................................................12

CHAPTER

1 INTRODUCTION .............................................................................................................13

2 LITERATURE REVIEW ..................................................................................................17

2.1 Experimental Research ................................................................................................172.2 Analytical Research .....................................................................................................18

3 COUPLED BARGE COLLISION ANALYSIS................................................................20

3.1 Introduction .................................................................................................................203.1.1 Barge Loading and Unloading Behavior 20......................................................3.1.2 Coupled Analysis Algorithm 21........................................................................3.1.3 Use of Experimental Data for Coupled Analysis Validation 21........................

3.2 Barge Impact Test Cases Selected for Validation: Case 1 and Case 2 .......................223.3 Software Selection and Model Development ..............................................................22

3.3.1 Coupled Analysis Module Parameters..........................................................243.3.2 Accounting for Payload Sliding During Impact Testing 25..............................

3.4 Comparison of Analytical and Experimental Data ......................................................263.4.1 Case 1............................................................................................................263.4.2 Case 2 27............................................................................................................

4 SIMPLIFIED MULTIPLE-PIER COUPLED ANALYSIS...............................................37

4.1 Overview......................................................................................................................374.2 Linearized Barge Force-Crush Relationship................................................................374.3 Reduction of the Bridge Model....................................................................................38

4.3.1 Uncoupled Condensed Stiffness Matrix 39.......................................................4.3.2 Lumped Mass Approximation 41......................................................................

4.4 Multiple-Pier Coupled Analysis Simplification Algorithm.........................................42

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5 SIMPLIFIED-COUPLED ANALYSIS DEMONSTRATION CASES ............................47

5.1 Introduction..................................................................................................................475.2 Geographical Information, Structural Configuration, and Impact Conditions ............48

5.2.1 Case 3 48............................................................................................................5.2.2 Case 4 49............................................................................................................5.2.3 Case 5 50............................................................................................................

5.3 Comparison of Simplified and Full-Resolution Results ..............................................515.4 Conclusions from Simplified-Coupled Analysis Demonstrations...............................525.5 Dynamic Amplification of the Impacted Pier Column Internal Forces.......................53

6 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH ............62

6.1 Conclusions..................................................................................................................626.2 Recommendations for Future Research .......................................................................63

APPENDIX

A SUPPLEMENTARY COUPLED ANALYSIS VALIDATION DATA ...........................64

B CONDENSED UNCOUPLED STIFFNESS MATRIX CALCULATIONS ....................72

C SIMPLIFIED-COUPLED ANALYSIS CASE OUTPUT.................................................76

D ENERGY EQUIVALENT AASHTO IMPACT CALCULATIONS................................95

REFERENCES ............................................................................................................................100

BIOGRAPHICAL SKETCH .......................................................................................................102

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LIST OF TABLES

Table page 1-1 Case descriptions: use, configuration, and impact data ....................................................16

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LIST OF FIGURES

Figure page

3-1 Coupling between barge and bridge (after Consolazio and Cowan 2005) ........................29

3-2 Stages of barge crush (after Consolazio and Cowan 2005)................................................30

3-3 Structural configurations analyzed (not to relative scale) ..................................................31

3-4 SDF barge force-crush relationship derived from experimental and analytical data .........32

3-5 Sliding criterion between payload and barge.....................................................................33

3-6 Comparison of Case 1 coupled analysis output and P1T4 experimental data ...................34

3-7 Comparison of Case 2 coupled analysis output and B3T4 experimental data...................35

3-8 Comparison of Case 2 coupled analysis output and B3T4 experimental data: Impulse...............................................................................................................................36

4-1 Derived and AASHTO SDF barge force-crush relationships (unloading curves not shown)............................................................................................43

4-2 Plan view of multiple pier numerical model and location of uncoupled springs in two-span single-pier model............................................................................................44

4-3 Structural configuration analyzed in Case 3 ......................................................................45

4-4 Plan view of multiple pier numerical model and location of lumped masses in two-span single-pier model............................................................................................46



5-3 Comparison of Case 3 simplified and full-resolution coupled analyses............................57



5-6 Time computation comparison of coupled analyses..........................................................60

5-7 Comparison of demonstration case simplified, full-resolution, and static analyses ..........61

A-1 Analytical output comparison to experimental P1T4 barge impact data...........................65


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A-5 Analytical output comparison to experimental B3T2 barge impact data ..........................69



C-1 Case 3 AASHTO curve coupled analysis output comparison at impact location..............77

C-2 Case 3 AASHTO curve coupled analysis output comparison at pier column top .............78

C-3 Case 3 AASHTO curve coupled analysis output comparison at pile head........................79







C-10 Case 3 bilinear curve coupled analysis output comparison at impact location..................86

C-11 Case 3 bilinear curve coupled analysis output comparison at pier column top .................87

C-12 Case 3 bilinear curve coupled analysis output comparison at pile head............................88







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LIST OF ABBREVIATIONS

L or L appended to symbol, indicates symbol exclusivity to left-flanking structure

R or R appended to symbol, indicates symbol exclusivity to right-flanking structure

[ ]F flexibility matrix

[ condensedK ] condensed stiffness matrix

couplingK off-diagonal (coupling) stiffness term

ΔK translational stiffness term

θK plan-view rotational stiffness term

unitM unit moment

Hm mass of half-span of superstructure

bm mass of barge

pm mass of payload

0u initial sliding velocity of payload

unitV unit shear force

couplingVθ shear due to coupled stiffness and plan-view rotation

ΔV shear due to translational stiffness and translation

pW weight of payload

Δ translation

MΔ translation due to unit moment

VΔ translation due to unit shear

μ static coefficient of friction between payload and barge

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θ plan-view rotation

Mθ plan-view rotation due to unit moment

Vθ plan-view rotation due to unit shear

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Abstract of Thesis Presented to the Graduate School of the University of Florida in Partial Fulfillment of the

Requirements for the Degree of Master of Science

SIMPLIFIED DYNAMIC BARGE COLLISION ANALYSIS FOR BRIDGE PIER DESIGN

By

Michael Thomas Davidson

August 2007

Chair: Gary R. Consolazio Cochair: Marc I. Hoit Major: Civil Engineering

The American Association of State and Highway Transportation Officials barge impact

provisions, pertaining to bridges spanning navigable waterways, utilize a static force approach to

determine structural demand on bridge piers. However, conclusions drawn from experimental

full-scale dynamic barge impact tests highlight the necessity of quantifying bridge pier demand

with consideration of additional forces generated from dynamic effects. Static quantification of

bridge pier demand due to barge impact ignores mass related inertial forces generated by the

superstructure which can amplify restraint of underlying pier columns.

An algorithm for efficiently performing coupled nonlinear dynamic barge impact analysis

on simplified bridge structure-soil finite element models is presented in this thesis. The term

“coupled” indicates the impact of a finite element bridge model and a respective single

degree-of-freedom barge model traveling at a specified initial velocity with a specified

force-deformation relationship. Coupled analysis is validated using experimental data. Also,

results from simplified and full-resolution analyses are compared for several cases to illustrate

robustness of the algorithm for various barge impact energies and pier types. Simplified coupled

dynamic analysis is shown to accurately capture dynamic forces and amplification effects.

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CHAPTER 1 INTRODUCTION

Potential loss of life and detrimental economic consequences due to bridge failure from

waterway vessel collision have been realized numerous times throughout modern history.

Catastrophic bridge failure events due to vessel collision, which occur approximately once a year

worldwide (Larsen 1993), led to the development of bridge design specifications for vessel

collision. The American Association of State and Highway Transportation Officials (AASHTO)

Guide Specification and Commentary for Vessel Collision Design of Highway Bridges is used

along with characteristics of a given waterway and the accompanying waterway traffic to

determine static design loads, which are applied to respective piers for impact design purposes

(AASHTO 1991). Even though the AASHTO specifications are used for bridge pier design due

to ship and barge collision, limited barge impact data was available for use in their development.

In April 2004, Consolazio et al. (2006) conducted full-scale experimental barge impact

testing on bridge piers of the Old St. George Island Causeway Bridge located in Apalachicola,

Florida. Key findings from the experiments that are pertinent to the research presented in this

thesis include:

• Inertial forces due to acceleration of bridge component masses can contribute significantly to overall pier response during a collision event;

• Significant portions of the impact load can transfer (or “shed”) into the superstructure; and,

• Superstructure resistance is comprised of displacement-dependent and mass-dependent (inertial) forces. Inertial forces can produce a momentary increase in pier restraint during initial impact stages, and amplify structural demand on pier columns.

Restraint of a bridge pier due to acceleration of the mass of the overlying superstructure, and the

corresponding amplification of forces developed in the pier columns during initial stages of

barge collision events, are not addressed in current static design procedures. In contrast,

dynamic time-history analysis of bridges inherently accounts for such amplification effects.

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However, due to the unique characteristics of each bridge, impact load time-histories vary from

bridge to bridge. Coupled dynamic analysis addresses this issue by employing a single

degree-of-freedom (SDF) barge mass, impact velocity, and vessel force-crush relationship to

simulate barge impact at a specified bridge pier location. This method enables efficient

time-history analysis that yields time-varying barge collision load and bridge response data

specific to each bridge structure. To validate the procedure, coupled analysis is performed and

compared with experimental data for single-pier and multiple-pier cases. A summary of all

analysis cases presented in this thesis is given in Table 1-1.

However, coupled full-resolution bridge finite element (FE) models are cumbersome to

analyze dynamically and time-history analysis of models of such size is not common in current

practice. To facilitate use of coupled analysis in design settings, simplifying modifications are

made to the barge and bridge structural models subject to impact. Specifically, to alleviate the

onus of developing an appropriate barge force-crush relationship for each of the possible barge

types, a simplified crush curve that is in accordance with current AASHTO design standards is

employed. Second, an algorithm is presented which incorporates coupled analysis but reduces a

multiple-pier model to essentially a pseudo-single pier model (with adjacent spans, springs, and

lumped masses) thereby significantly reducing required analysis time. Simplified-coupled

dynamic barge impact analysis is performed and compared to results from full-resolution models

for a range of bridge and collision configurations. In comparison to full bridge model coupled

time-history analysis, results from respective simplified models are sufficiently accurate for

design purposes. Comparisons are also made between static and dynamic analysis predictions of

bridge pier structural demand for each case. By employing coupled analysis with a simplified

crush curve and simplified bridge structural model, design-oriented software is produced that can

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efficiently quantify collision induced bridge pier demand, including capture of dynamic

amplification effects.

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Table. 1-1. Case descriptions: use, configuration, and impact data.

Barge impact parameters Case Use a No. Piers Spans Weight Velocity Energy 1 V 1 0 5.37 MN (604 T) 1.33 m/s (2.59 knots) 0.484 MN-m (357 kip-ft) 2 V 4 3 3.06 MN (344 T) 0.787 m/s (1.53 knots) 0.097 MN-m (71.3 kip-ft) 3 U/D 5 4 1.78 MN (200 T) 1.03 m/s (2.00 knots) 0.096 MN-m (70.9 kip-ft) 4 D 5 4 18.0 MN (2020 T) 1.54 m/s (3.00 knots) 2.18 MN-m (1610 kip-ft) 5 D 5 4 68.7 MN (7720 T) 3.63 m/s (7.00 knots) 46.0 MN-m (34000 kip-ft) a V = Validation; U = Uncoupled Condensed Stiffness Calculation; D = Demonstration

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CHAPTER 2 LITERATURE REVIEW

2.1 Experimental Research

In 1983, Meier-Dörnberg conducted reduced scale impact tests on barge bows using a

pendulum impact hammer. Static crush tests were also performed on reduced scale barge bows.

Results from this study were used to develop relationships between kinetic energy, barge bow

crush depth, and static impact force. These relationships comprise a major portion of the

collision-force calculation procedure adopted in the AASHTO specifications (1991). However,

this research did not address phenomena such as bridge superstructure effects and dynamic

amplification, nor did the tests involve pier or bridge response.

During this same time and afterward, full-scale experimental barge collision tests were

conducted in connection with the U.S. Army Corps of Engineers (USACE). In 1989, lock gate

impact tests were performed with a nine-barge flotilla traveling at low velocities at Lock and

Dam 26 near Alton, Illinois (Goble et al. 1990). In 1997, four-barge flotilla impact tests were

conducted on concrete lock walls at Old Lock and Dam 2, near Pittsburgh, Pennsylvania (Patev

et al. 2003). Additional lock wall tests were conducted with a fifteen-barge flotilla in 1998 at the

Robert C. Byrd Lock and Dam in West Virginia (Arroyo et al. 2003). All of these tests were

performed on lock walls and lock gates, which produce fundamentally different structural

responses to collision loading in comparison to that of bridge piers.

The impact testing (Consolazio et al. 2006) of the old St. George Island Bridge,

constructed in the 1960s, constitutes the only experimental research that explicitly measured

barge impact forces on bridge piers using full-scale tests. The experiments were divided into

three series of impact tests using a single barge and various pier/bridge structural configurations.

The first series (termed the P1 series) consisted of eight impacts on a single, stiff channel pier

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(termed Pier 1-S) by a loaded barge with an impact weight of 5.37 MN (604 T) and impact

velocities approaching 1.8 m/s (3.5 knots). The second series of tests (termed the B3 series)

consisted of four impacts on a multi-span, multi-pier partial bridge structure by an empty barge

with an impact weight of 3.06 MN (344 T) and impact velocities approaching 0.78 m/s

(1.5 knots). The third series (termed the P3 series) consisted of three impacts on a single,

flexible pier (termed Pier 3-S) by an empty barge with an impact weight of 3.06 MN (344 T) and

impact velocities approaching 0.95 m/s (1.8 knots). These tests form an important dataset for

validating barge collision analysis methods.

2.2 Analytical Research

Development and analysis of very high-resolution contact-impact FE models (those with

tens to hundreds of thousands of elements) that simulate nonlinear dynamic barge impact on

bridge piers have been feasible as a research tool for approximately a decade. In preparation for

the full-scale St. George Island experimental barge impact testing, high-resolution FE pier

models were developed to determine appropriate experimental conditions with respect to barge

impact velocity and safety (Consolazio et al. 2002). Reanalysis of the models using

experimental data complimented the research findings from the experimental program

(Consolazio et al. 2006).

High-resolution FE models of single-barges and multi-barge flotillas were analyzed when

pier columns of various shape and dimension were subject to a variety of barge impact

simulations (Yuan 2005). These analytical results were used to develop a set of empirical

formulas for barge impact force quantification as an improvement to the current static design

procedures. Also, high-resolution FE single-barge models were developed and subjected to

quasi-static loading by various stiff impactors in an effort to better quantify barge force-crush

relationships (Consolazio and Cowan 2003).

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As an alternative to very high resolution contact-impact FE analysis, coupled barge-pier

analysis was developed (Consolazio et al. 2004a, Consolazio et al. 2004b). Coupled analysis

simulates a SDF barge model (with specified mass, velocity, and force-crush relationship)

colliding with a multiple degree-of-freedom (MDF) bridge-pier-soil model. The coupled

analysis required the use of a barge force-crush relationship, which was developed for a common

barge type using high-resolution FE models. The force-crush curves encompass loading and

unloading behavior derived from quasi-static cyclic loading (Consolazio and Cowan 2005).

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CHAPTER 3 COUPLED BARGE COLLISION ANALYSIS

3.1 Introduction

Within the context of coupled analysis, the term “coupled” refers to the use of a shared

contact force between the barge and impacted bridge structure (Fig. 3-1). The impacting barge is

assigned a mass, initial velocity, and bow force-crush relationship. Traveling at the prescribed

initial velocity, the barge impacts a specified location on the bridge structure and generates a

time-varying impact force in accordance with the force-crush relationship of the barge and the

relative displacements of the barge and bridge model at the impact location. The barge is

represented by a SDF model, and the pier structural configurations and soil parameters of the

impacted bridge structure constitute a MDF model. The MDF pier-soil model, subject to the

shared time-varying impact force, displaces, develops internal forces, and interacts with the SDF

barge model through the shared impact force during the analysis. Hence, coupled analysis

automatically generates the barge impact load time-history specific to each bridge structural

configuration and impacting barge type. This overcomes the challenge of pre-quantifying the

time-varying barge impact load as a necessary component of time-history analysis.

3.1.1 Barge Loading and Unloading Behavior

Barge behavior is represented by a force-crush relationship, consisting of a loading curve,

unloading curves, and a specified yield point (Fig. 3-2). The yield point represents the crush

depth beyond which plastic deformations occur. Any subsequent unloading beyond this point is

determined according to the specified unloading curves. Until the crush depth corresponding to

yield is reached, loading and unloading occurs elastically along the specified curve (Fig. 3-2A).

A series of unloading curves represent the unloading behavior at various attained maximum

crush depths (Fig. 3-2B). After unloading, if the barge is no longer in contact with the pier, no

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impact force is generated (Fig. 3-2C). Alternatively, if reloading occurs (Fig. 3-2D), it is

assumed to occur along the previously traveled unloading curve. Plastic deformation subsequent

to complete reloading occurs along the originally specified loading curve (Fig. 3-2D).

Additional details of this model are given in Consolazio and Cowan (2005).

3.1.2 Coupled Analysis Algorithm

Algorithmically, the coupled analysis procedure involves a SDF barge code interacting

with a separate nonlinear dynamic pier-soil analysis code at a specified node of the MDF

pier-soil model. Specifically, coupled analysis utilizes an explicit time-step barge impact force

determination procedure and links the output, the resulting impact force, with a respective

numerical MDF pier-soil model analysis code (Hendrix 2003). The pier-soil analysis code then

responds to the impact force by generating iterative displacements and forces throughout the

MDF model.

3.1.3 Use of Experimental Data for Coupled Analysis Validation

Coupled analysis was previously developed and demonstrated as a proof-of-concept

using analytical data (Consolazio and Cowan 2005). Output from very high-resolution FE

models consisting of a MDF impacting barge and a MDF impacted pier were compared to output

obtained from coupled analysis of a SDF barge and MDF pier model. At present, experimental

data is now available for validation of the coupled analysis procedure. Using data from the

full-scale barge impact experiments (Consolazio et al. 2006), validation of the coupled analysis

procedure is carried out in four stages: select appropriate pier structures from the experimental

dataset; develop respective models in a nonlinear dynamic finite element analysis (NDFEA) code

capable of conducting coupled analysis; analyze the models using respective barge impact

conditions and coupled analysis; and, compare time-history results from the coupled analysis to

those obtained experimentally.

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3.2 Barge Impact Test Cases Selected for Validation: Case 1 and Case 2

Data was collected more extensively from Pier 1-S than from any other pier in the 2004

full-scale experimental test set (Consolazio et al. 2006). Furthermore, a single pier is

representative of the type of structure often used in static design procedures for barge collision

analysis (Knott and Prucz 2000). Hence, a single pier (Pier 1-S) was selected for coupled

analysis validation using experimental data (Fig. 3-3A). Of the eight experimental tests

conducted on Pier 1-S, the fourth test (termed P1T4) consisted of a head-on impact at an

undamaged portion of the barge bow, as would be assumed in bridge design. Test P1T4, with

velocity and impact weight as specified in Table 1-1, was selected for Case 1.

In addition to validating the coupled analysis procedure for a single-pier, data from the

partial bridge (B3 series) tests were employed for validation purposes. Regarding impact

conditions used for validation, the fourth test (termed B3T4) generated the largest pier response

among the B3 test series. Hence, test B3T4, with velocity and impact weight as specified in

Table 1-1, was selected for Case 2 (Fig. 3-3B).

3.3 Software Selection and Model Development

Coupled analysis was previously implemented in the commercial pier analysis software,

FB-Pier (2003), and was shown to produce force and displacement time-histories in agreement

with those obtained from high-resolution contact-impact FE pier-soil model simulations.

Subsequent to implementation of the coupled analysis procedure in FB-Pier, an enhanced

package called FB-MultiPier (2007) was released. FB-MultiPier possesses the same analysis

capabilities as FB-Pier (including coupled analysis) but also has the ability to analyze bridge

structures containing superstructure elements. Therefore, FB-MultiPier was selected for all

model development and analysis conducted in this study.

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FB-MultiPier employs fiber-based frame elements for piles, pier columns, and pier caps;

flat shell elements for pile caps; beam elements, based on gross section properties, for

superstructure spans; and, distributed nonlinear springs to represent soil stiffness. Transfer

beams transmit load from bearings, for which the stiffness and location are user-specified, to the

superstructure elements. FB-MultiPier permits Rayleigh damping, which was applied to all

structural elements in the models used for this study such that approximately 5% of critical

damping was achieved over the first five natural modes of vibration.

FB-MultiPier allows either linear elastic or material-nonlinear analysis of structural

elements. Linear elastic analysis was selected for all structural (non-soil) element components of

models used in this study. This approach was taken because the 2004 full-scale barge impact

experiments were non-destructive (Consolazio et al. 2006) and post-test inspection of the pier

structures subjected to collision loading indicated that the structural components had remained

largely in the elastic range.

Structural models of Case 1 and Case 2 (Fig. 3-3A and Fig. 3-3B, respectively) were

developed from original construction drawings and direct site investigation measurements. The

Case 2 structural model was limited to four piers, with springs representing the stiffness

contributions of piers beyond Pier 5-S (Fig. 3-3B), as contribution to structural response from

these piers was expectedly small (Consolazio et al. 2006). The soil model spring system for

Case 1 was developed based on boring logs and dynamic soil properties obtained from a

geotechnical investigation conducted in parallel with the 2004 full-scale barge impact testing

(McVay et al. 2005). For the development of the Case 2 soil-spring system, boring logs formed

the sole data source available.

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For each model, a preliminary analysis was conducted in which the experimentally

measured time-history load was directly applied at the impact point for the specified test case.

The resulting displacement time-history of the structure was then compared to the experimentally

measured displacement time-history at the impact point. Output from the direct analysis and

comparison to experimental data aided in calibration of each model. Consequently, because

analytical application of the experimentally measured load time-history was shown to produce

pier response in agreement with that of the experimental data, the direct analysis comparison

provided a baseline means of judging the efficacy of the coupled analysis procedure.

3.3.1 Coupled Analysis Module Parameters

Within the coupled analysis procedure, the barge is modeled by a SDF point mass and

nonlinear compression spring. Barge impact conditions for the validation cases (P1T4 and

B3T4) were directly measured during the experimental tests. Thus, the experimental impact

weights and velocities were directly input into analytical Case 1 as 5.37 MN (604 T) traveling at

1.33 m/s (2.59 knots) and Case 2 as 3.06 MN (344 T) traveling at 0.79 m/s (1.53 knots),

respectively.

The loading portion of the barge force-crush relationship used for Case 1 and Case 2

(Fig. 3-4) was developed from impact-point force and displacement time-history data measured

during the P1T4 test; P1T4 was selected because of the undamaged bow impact location and

head-on nature of the collision event. The portion of the barge force-crush relationship up to the

peak force was obtained by performing coupled analysis using P1T4 impact conditions, and an

initially arbitrary force-crush relationship. After analysis completion, the coupled analysis

prediction of impact force was compared to that experimentally measured during the P1T4 test.

The analytical force-crush relationship was then adjusted to more closely match that measured

experimentally. After several iterations of this calibration process, a force-crush loading

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relationship was obtained that produced force time-history data in agreement with the

experimental measurements of impact force.

The experimentally derived loading portion of the force-crush curve (Fig. 3-4) has a peak

impact force value of 5.74 MN (1065 kips) at a crush depth of 12.07 cm (4.75 in). Explicit

derivation of forces beyond this point, pertaining to the barge-bow impact force-crush

relationship, was not possible using the experimental dataset. However, barge bow force-crush

data are available in the literature that apply to the shape of the impacted pier in the P1T4 test;

specifically, a rectangular (flat) surface impactor. This data was obtained by subjecting a

high-resolution FE barge model to quasi-static crushing by square (flat) 1.8 m (6 ft) and 2.4 m

(8 ft) impactors (Consolazio and Cowan 2003). In the present study, barge force-crush

parameters pertaining to crush depths beyond that corresponding to the peak force were

proportioned from high-resolution FE force-crush data. Specifically, these parameters are: the

yield point, structural softening beyond the peak force, and the force plateau level beyond

softening (Fig. 3-4). The unloading curves (Fig. 3-4) chosen for Case 1 and Case 2 exhibit

steeper unloading paths at smaller crush-depths and shallower unloading paths at larger crush

depths. The unloading curves are consistent, with respect to qualitative shape, with those

employed in a prior study for a common barge type subject to quasi-static crush by square piers

(Consolazio and Cowan 2005).

3.3.2 Accounting for Payload Sliding During Impact Testing

During the Pier 1-S test series, payload in the form of 16.76 m (55 ft) reinforced concrete

bridge superstructure span segments was placed on the barge to simulate a loaded impact

condition. However, the payload was observed to slide during the collision events, implying the

development of frictional forces and dissipation of energy (Consolazio et al. 2006). In general

bridge design, the payload would not be assumed to slide. However, for the purpose of

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validating the coupled analysis procedure as accurately as possible, enhancements were made to

the pre-existing coupled analysis procedure to numerically account for payload sliding (Fig. 3-5).

At each time-step and iteration, the ratio of barge acceleration (which, before sliding occurs, is

equal to the payload acceleration) to gravitational acceleration was computed and compared to

the static coefficient of friction (μ ) between the barge and the payload. When the acceleration

ratio exceeded the static coefficient of friction, sliding was initiated (Fig. 3-5B). At sliding

initiation, the barge payload was assigned an initial velocity ( ) relative to the underlying

barge, equal to the corresponding current velocity of the barge-payload system. The payload was

assumed to continue sliding until the initial payload kinetic energy was completely dissipated

through friction. At all points in time during which sliding occurred, a constant frictional force,

equal to the product of the static coefficient of friction and the weight of the payload ( ), was

applied to the barge. When the sliding kinetic energy of the payload barge was dissipated, the

payload mass ( ) and barge mass ( ) were assumed to rejoin as a single loaded

barge-payload system, as before sliding (Fig.

0u

pW

pm bm

3-5A). For the P1T4 test, a sliding distance of

0.376 m (14.8 in) was predicted from the module modifications, which agreed very well with the

observed payload slide of approximately 0.38 m (15 in).

3.4 Comparison of Analytical and Experimental Data

3.4.1 Case 1

The Case 1 impact load time-history (Fig. 3-6A) is nearly identical to the respective

experimental curve up to the peak load, and expectedly so, because the portion of the barge

force-crush relationship (Fig. 3-4), up to the peak impact load, was derived from the impact force

and displacement data acquired during the Case 1 (P1T4) collision event. Additionally, the

analytical and experimental agreement for portions of the Case 1 force time-history curve

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beyond the peak justifies the assumptions made during the development of the load softening,

load plateau, and unloading components of the force-crush curve (Fig. 3-4).

The analytically determined peak value of pier displacement exceeds the experimental

value by 16% (Fig. 3-6B). Supplementary coupled analyses of the Pier 1-S model were

conducted with impact velocities measured during similar and higher impact-energy P1 series

tests. Comparisons of displacement output from these analyses (Appendix A) to respective

experimental data show discrepancies of comparable or lesser magnitude to those of Case 1.

3.4.2 Case 2

Case 2, in direct contrast to Case 1, consists of a low-energy barge collision event on a

flexible pier with superstructure restraint. Case 1 and Case 2 share only the barge force-crush

relationship derived from the P1T4 experimental data. The Case 2 experimental and analytical

force time-histories (Fig. 3-7A) embody similar qualitative shapes; however, the analytical peak

force magnitude is larger than the experimental counterpart. Despite the disparity in magnitude,

numerical integration of the curves indicates that the shape and magnitude of the impulse, as a

function of time, agree well between the experimental and analytical results (Fig. 3-8). This

implies that the change in momentum of the barge was accurately predicted by the coupled

analysis and produced a pier response similar to that measured experimentally.

The concord of the analytical and experimental time-history of displacement (Fig. 3-7B)

demonstrates the proficiency of the coupled analysis procedure in adequately predicting barge

collision response for piers of varying stiffness. Accurate pier response predictions are

maintained while incorporating superstructure effects. Agreement of pier response is the most

important outcome of the coupled analysis procedure, as the accompanying internal forces

generated throughout the MDF pier-soil model ultimately govern the pier structural member

design. The coupled analysis procedure effectively shifts the analytical focus away from

27

determination of the barge impact force, and centers the emphasis on determining pier structural

demand.

Coupled analysis also inherently captures dynamic phenomena exhibited during

barge-bridge collisions. As evidenced by the time-history plots of Case 1 and Case 2 (Fig. 3-6

and Fig. 3-7), the peak impact force and displacement do not occur simultaneously for individual

experimental test cases involving appreciable impact-energies (Consolazio et al. 2006). Static

procedures do not account for peak load-displacement time disparity or the potential

amplification effects intrinsic to the early stages of collision events for bridge structures.

Coupled analysis automatically accounts for these effects.

28

FF

Pier structure

Soilstiffness

Crushable bowsection of barge

Barge

Barge and bridge modelsare coupled together througha common contact force

SDF barge model MDF bridge model

Bridge motionBarge

motion

superstructure

Figure 3-1. Coupling between barge and bridge (after Consolazio and Cowan 2005).

29

ImpactForce

CrushDepth

Yieldpoint

Elastic loading/unloading

Loading curve

A

ImpactForce

CrushDepth

Unloading curve

Initiationof unloading

B

ImpactForce

CrushDepth

Barge and bridge not in contact

C

ImpactForce

CrushDepth

Plastic loading occurs along loading curve

Reloading occurs along same path

as unloading

D

Figure 3-2. Stages of barge crush (after Consolazio and Cowan 2005). A) Loading. B) Unloading. C) Barge not in contact with pier. D) Reloading and continued plastic deformation.

30

Impact

Pier 1-S

A

Impact

Pier 2-S Pier 3-S Pier 4-S Pier 5-S

Springs modelingadditional spansbeyond Pier 5-S

B

Figure 3-3. Structural configurations analyzed (not to relative scale). A) Case 1: Single pier. B) Case 2: Four piers with superstructure.

31

Crushable barge bow

SDF Barge MDF Pier

Crush Distance (mm)

Crush Distance (in)

Impa

ct F

orce

(MN

)

Impa

ct F

orce

(kip

s)

0 50 100 150 200 250 300

0 2 4 6 8 10

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

5.5

0

150

300

450

600

750

900

1050

1200Loading CurveUnloading Curves

Figure 3-4. SDF barge force-crush relationship derived from experimental and analytical data.

32

F

Barge

Payload

Total barge-payload masscontributes to impact force

No relative motion

Gravitational acceleration< Static coefficient of friction

between barge and payloadBarge acceleration

Wp

μ

m p

m b

A

F

Barge Barge mass and constant payload frictional force contribute to impact force

u 0Payload

μWp

Gravitational acceleration> Static coefficient of friction

between barge and payloadBarge acceleration

m p

m b

Wp

B

Figure 3-5. Sliding criterion between payload and barge. A) No sliding. B) Sliding.

33

Time (s)

Impa

ct F

orce

(MN

)

Impa

ct F

orce

(kip

s)

0 0.25 0.5 0.75 1 1.25 1.50

1

2

3

4

5

0

200

400

600

800

1000Coupled Analysis OutputExperimental Data

A

Time (s)

Pier

Disp

lace

men

t (m

m)

Pier

Dis p

lace

men

t (in

)

0 0.25 0.5 0.75 1 1.25 1.5-5

-2.5

0

2.5

5

7.5

10

12.5

15

-0.10

0.00

0.10

0.20

0.30

0.40

0.50Coupled Analysis OutputExperimental Data

B

Figure 3-6. Comparison of Case 1 coupled analysis output and P1T4 experimental data. A) Impact force. B) Pier displacement.

34

Time (s)

Impa

ct F

orce

(MN

)

Impa

ct F

orce

(kip

s)

0 0.25 0.5 0.75 1 1.25 1.50

0.5

1

1.5

2

0

100

200

300

400Coupled Analysis OutputExperimental Data

A

Time (s)

Pier

Disp

lace

men

t (m

m)

Pier

Disp

lace

men

t (in

)

0 0.25 0.5 0.75 1 1.25 1.5-20

-10

0

10

20

30

40

50

-0.40

0.00

0.40

0.80

1.20

1.60Coupled Analysis OutputExperimental Data

B

Figure 3-7. Comparison of Case 2 coupled analysis output and B3T4 experimental data. A) Impact force. B) Pier displacement.

35

Time (s)

Impu

lse (M

N-s

ec)

Impu

lse (k

ip-s

ec)

0 0.25 0.5 0.75 1 1.25 1.50

0.1

0.2

0.3

0.4

0

20

40

60

80

Coupled Analysis OutputExperimental Data

Figure 3-8. Comparison of Case 2 coupled analysis output and B3T4 experimental data: Impulse.

36

CHAPTER 4 SIMPLIFIED MULTIPLE-PIER COUPLED ANALYSIS

4.1 Overview

At current computer processing speeds, barge impact time-history analysis of bridge

models can require between tens of minutes to several hours of processing time. Two

simplifications may be applied to the coupled analysis of bridge structural models to reduce

analysis time and facilitate its use in design settings. First, a simplified alternative to the

experimentally and analytically derived crush curve may be used in design when more detailed

barge force-crush behavior is not available. The bilinear curve found in the current static

AASHTO design specifications (Fig. 4-1) may be used for general barge-bridge collision design

applications. Second, multiple-pier models may be reduced to a pseudo-single pier model (with

two attached superstructure spans) and analyzed to produce results that match to a satisfactory

degree of accuracy, those obtained from corresponding full-resolution (multi-span, multi-pier)

models.

4.2 Linearized Barge Force-Crush Relationship

The nonlinear loading portion of the barge force-crush curve, developed from P1T4

experimental data (Fig. 4-1), is specific to the barge used in the 2004 impact experiments.

Phenomena such as structural-softening beyond the peak force level for each combination of

barge type and impactor shape are not well documented in the literature and further study is

warranted before these components of barge bow crushing behavior may be quantified for

general application. Hence, the use of a simple bilinear force-crush relationship, such as that

found in the AASHTO barge-collision specifications, is desirable at present as long as such a

curve produces reasonable results.

37

The AASHTO force-crush relationship is in reasonable agreement with the P1T4

experimentally determined based force-crush curve. The crush depth at which the AASHTO and

experimental curves shift from the initial linear portion to the subsequent linear portion occur at

103.63 mm (4.08 in) and 120.65 mm (4.75 in), respectively. For convenience, these locations

are termed the shift points. Note that the AASHTO force corresponding to the shift point,

6.17 MN (1386 kips), is significantly greater than that found in the experimentally based curve,

4.74 MN (1065 kips). Additionally, the AASHTO curve exhibits positive stiffness regardless of

crush depth, whereas the curve employed in the validation of the coupled analysis method is

assumed to exhibit perfectly plastic behavior at high crush depths (Fig. 4-1). Consequently, the

AASHTO curve yields higher impact forces than the experimental data for all barge crush

depths, and is therefore conservative.

4.3 Reduction of the Bridge Model

Barges impart predominantly horizontal forces to impacted bridge piers during collision

events. Displacement and acceleration based superstructure restraint (due to superstructure

stiffness and mass, respectively) can attract a significant portion of the horizontal forces and

cause the impact load to “shed” to the superstructure (Consolazio et al. 2006). The horizontal

force shed to the superstructure then propagates (initially) away from the impacted pier.

Consequently, lateral translational and plan-view rotational stiffnesses influence the structural

response as the force propagates through the superstructure from the impacted pier to adjacent

piers. Simultaneously, the distributed mass of the superstructure alternates between a source of

inertial resistance to a source of inertial load that respectively restrains or must be absorbed by

other portions of the bridge structure. Simplification of the multiple-pier structural model,

therefore, must adequately retain the influence of adjacent non-impacted (the lateral translational

38

and plan-view rotational stiffnesses of the adjacent piers; and, the dynamically participating mass

of the superstructure).

4.3.1 Uncoupled Condensed Stiffness Matrix

The stiffness DOF of a bridge model, beyond the superstructure spans that extend from

the impacted pier (Fig. 4-2), may be approximated by equivalent lateral translational springs and

plan-view rotational springs. These springs are linear elastic and represent the predominant DOF

of the linear elastic structural elements in the full-resolution model at piers adjacent to the

impacted pier. Soil nonlinearities at piers other than the impacted pier are ignored during

formation of the translational and rotational springs.

Replacement of numerous DOF from the flanking portions of a full bridge model

(Fig. 4-2) by two uncoupled springs at each end of a simplified two-span single-pier model may

be described in terms of a condensed stiffness matrix:

[ ] ⎥⎦

⎤⎢⎣

⎡= Δ

θKKKK

Kcoupling

couplingcondensed (4.1)

where [ is the condensed stiffness matrix of the flanking bridge portion eliminated at

each side of the impacted pier; is the condensed lateral translational stiffness term;

is an off-diagonal stiffness term that couples the translational DOF to the rotational DOF; and

is the condensed stiffness plan-view rotational stiffness term. In the simplified model, the

diagonal terms and are represented by translational and rotational springs, respectively,

and the terms are neglected. The exclusion of in the simplified model is

justified by examining the forces generated by the condensed stiffness terms on one side of an

example five-pier model.

]condensedK

ΔK couplingK

θK

ΔK θK

couplingK couplingK

39

A channel pier was added to the previously discussed four-pier Case 2 model, using

bridge plans of the old St. George Island Bridge. This new five-pier model (Fig. 4-3) is referred

to as Case 3, as defined in Table 1-1. Through flexibility inversion (Fig. 4-2), the left-flanking

bridge structure in Case 3 (consisting of Pier 1-S to Pier 2-S), is reduced to the 2-DOF linear

elastic condensed stiffness matrix in Eq. (4.1), where =ΔK 97.0 MN/m (554 kip/in);

2.58E+05 MN-m/rad (2.28E+09 kip-in/rad); and, =θK =couplingK 398 MN/rad

(8.95+04 kip/rad). In row one of , the term may be interpreted as a horizontal

shear force generated when a unit rotation (1 rad) is induced at the right-most node of the

left-flanking structure. Static application of a load of 1.84 MN (414 kips) to the central pier of

the Case 3 five-pier model induces a plan-view rotation of

[ ]condensedK couplingK

=θ 6.35E-06 rad at the location of

the condensed stiffness. The horizontal shear produced as a result of this rotation is:

θθ couplingcoupling KV = (4.2)

where is the shear produced from the coupling of rotational and translational DOF. In

this instance, 2.53E-03 MN (0.568 kips). In comparison, the horizontal shear produced

as a result of diagonal lateral stiffness

couplingVθ

=couplingVθ

=ΔK 97.0 MN/m (554 kip/in) and lateral displacement

4.62 mm (0.182 in) is: =Δ

Δ= ΔΔ KV (4.3)

where is the shear produced directly from lateral translation. For this loading,

0.448 MN (101 kips).

ΔV

=ΔV

The amount of horizontal shear generated at the location of the condensed stiffness

matrix, due to the coupling stiffness term, is very small relative to the amount of horizontal shear

40

generated due to the diagonal stiffness term ( is only 0.6% of ). A similar examination

of the and terms yields ratios of comparable values (

couplingVθ ΔV

θK couplingK Appendix B). The large

difference in magnitude between the two shear forces demonstrates that the off-diagonal stiffness

terms of generate negligible forces relative to those generated by the diagonal

stiffness terms. Uncoupling the condensed stiffness terms by applying two independent springs

is therefore warranted for design applications, as the uncoupled springs form a reasonable static

approximation of the stiffness of the excluded portions of the model.

[ condensedK ]

As a further simplification to the full-bridge model, the diagonal stiffness terms and

may be approximated by direct inversion of the individual diagonal flexibility coefficients.

Specifically, this entails directly inverting the translational

ΔK

θK

VΔ and rotational Mθ displacements,

respectively, induced by the application of a unit shear force and unit force-couple

on the applicable flanking structure (Fig.

unitV unitM

4-2

) and is simpler to carry out.

). This approximation produces only nominally

different magnitudes of stiffness with respect to that obtained by a flexibility matrix inversion

(Appendix B

If significant nonlinear behavior is expected at non-impacted piers, then loads

representative of the forces that will be shed to the superstructure, and subsequently transmitted

into these piers, should be used to compute displacements (flexibility coefficients). Inversion of

flexibility coefficients formed in this manner yields a condensed secant stiffness that may then be

employed in the simplified model as described previously.

4.3.2 Lumped Mass Approximation

Mass is attributed to each node of the NDFEA models in this study, which consequently,

approximate a distributed mass system under dynamic loading. Therefore, a portion of mass of

the excluded structural components is assumed to contribute to the structural response of the

41

simplified models. This mass is assumed to fall within the tributary area (Fig. 4-4) extending

along the spans beyond the piers adjacent to the impacted pier of a given full-resolution model.

The mass is lumped and placed at respective ends of the simplified model. The lumped mass

simplification is combined with the stiffness approximation (Fig. 4-2) to complete the simplified

two-span single-pier model.

4.4 Multiple-Pier Coupled Analysis Simplification Algorithm

Simplified coupled analysis occurs in two stages. First, the two-span single-pier model is

assembled by replacing extraneous portions of the multiple-pier model with uncoupled linear

elastic springs and half-span lumped tributary masses. Coupled analysis is then performed, as

previously discussed, with the AASHTO bilinear crush-curve being employed for the barge.

The simplification algorithm automatically retains the ability to capture dynamic effects,

such as amplification, not addressed in static procedures. Furthermore, hundreds to thousands of

DOF are eliminated because the non-impacted piers and respective superstructure spans from the

full-resolution model are omitted from the model.

42

Crush Distance (mm)

Crush Distance (in)

Impa

ct F

orce

(MN

)

Impa

ct F

orce

(kip

s)

0 50 100 150 200 250 300

0 2 4 6 8 10

0

1

2

3

4

5

6

7

0

300

600

900

1200

1500

Derived from experimental dataAASHTO

Figure 4-1. Derived and AASHTO SDF barge force-crush relationships (unloading curves not shown).

43

K =F =

Impact location on full bridge modelForm left-flanking and right-flanking structures, excluding impacted pier P-4 and the two connecting spans

P-1 P-2 P-3 P-4 P-5 P-6 P-7

Apply unit shear force at center of P-3 pile cap and center of P-5 pile cap

Vunit

P-1 P-2 P-3 P-5 P-6 P-7

Record shear-induced translations and rotations at center of P-3 pile cap and center of P-5 pile cap

Apply unit moment at center of P-3 pile cap and center of P-5 pile cap

M unit

P-1 P-2 P-3 P-5 P-6 P-7

M unit

Record moment-induced rotations and translations at center of P-3 pile cap and center of P-5 pile cap

Form 2x2 left-flanking and right-flanking flexibility matrices using displacements and invert to form condensed stiffness

Left-flanking structure Right-flanking structureP-1 P-2 P-3 P-5 P-6 P-7

ΔVLP-1 P-2

P-3

P-7P-6

P-5

ΔVR

Vunit

θVL θVR

P-4

Neglect off-diagonal stiffness and replace flanking-structures in full bridge model with diagonal stiffness as uncoupled springs

Impact location on two-span single-pier model

ΔVL

θML K couplingL K θ

L

L K LK ΔL

coupling

0 K θL

L 0K ΔL

K θR

LK ΔR

0

0LK ΔL

K θL

LK ΔR

K θR

L Lcondensed

-1

-1

K =F =K coupling

R K θR

L K RK ΔR

couplingR R

condensed

-1

-1

ΔMR

θMR

P-7P-6

P-5

P-1 P-2 P-3

ΔML

θML

ΔML

θVL

ΔVR

θMR

ΔMR

θVR

Figure 4-2. Plan view of multiple pier numerical model and location of uncoupled springs in two-span single-pier model.

44

Pier 1-S Pier 2-S Pier 3-S Pier 4-S Pier 5-S

Impact

Figure 4-3. Structural configuration analyzed in Case 3.

45

Impact location on full bridge modelForm left-flanking and right-flanking structures, excluding impacted pier P-4 and the two connecting spans

P-1 P-2 P-3 P-4 P-5 P-6 P-7

Calculate mass of half-span beyond P-3

m HL

P-1 P-2 P-3 P-5 P-6 P-7

P-4

Impact location on two-span single-pier model

Calculate mass of half-span beyond P-5

Form lumped mass equal to m HL Form lumped mass equal to m HR

Apply lumped masses in place of flanking-structure masses in full bridge model

Left-flanking structure Right-flanking structure

P-1 P-2 P-3 P-5 P-6 P-7

m HR

m HL m HR

m HL m HR

Figure 4-4. Plan view of multiple pier numerical model and location of lumped masses in two-span single-pier mode.

46

CHAPTER SIMPLIFIED-COUPLED ANALYSIS DEMONSTRATION CASES

5.1 Introduction

To illustrate the efficacy of the simplification algorithm, three demonstration cases

(FB-MultiPier bridge models) are presented. Each model was developed using methods

representative of those employed by bridge designers. Impact conditions prescribed for the

models are such that the range of scenarios encountered in practical bridge design for barge

impact loading is well represented. The cases employ the AASHTO bilinear barge crush-curve,

consist of impacted pier models of increasing impact resistance, and are subjected to impacts

with corresponding increases in impact energy. Time-history output of internal pier structural

member forces obtained from both full-resolution and simplified models are subsequently

compared for each case.

Each full-resolution model contains five piers: a centrally located impact pier and

additional structural components (soil, non-impacted piers, and superstructure spans) for a length

of two spans to either side of the central pier. A five-pier model contains a sufficient number of

piers and spans such that inclusion of additional piers would increase analytical computation

costs without appreciably improving the computed structural response. The appropriateness of

the decision to limit the full-resolution models to five piers is substantiated by the consistently

negligible acceleration response exhibited by the outer-most piers included in the five-pier

models. Alternatively stated, the added restraint provided by including additional piers is not

necessary, as the outer-most piers of the five-pier models are only nominally active throughout

the barge impact analysis.

47

A single time-step increment, 0.0025 sec, was employed for all demonstration analyses.

Each model also utilized Rayleigh damping, which is configured such that the first five vibration

modes undergo damping at approximately 5% of critical damping.

5.2 Geographical Information, Structural Configuration, and Impact Conditions

5.2.1 Case 3

The first demonstration case consists of analysis of the previously described Case 3

model (Fig. 4-3). This model was based on the old St. George Island Bridge from the

Apalachicola Bay area, linking St. George Island to mainland Florida, in the southeastern United

States. Apalachicola Bay is located approximately 80.5 km (50 mi) southwest of Tallahassee,

Florida in the “panhandle” portion of the state.

The structure of the old St. George Island Bridge, constructed in the 1960s, was detailed

in a prior report (Consolazio et al. 2006). Pertinent to demonstration Case 3, the superstructure

spanning from Pier 2-S to Pier 5-S (Fig. 4-3) consisted of 23 m (75.5 ft) concrete girder-and-slab

segments overlying concrete piers with waterline footings. Spanning the navigation channel and

one additional pier to either side, a 189 m (619.5 ft) continuous three-span steel girder and

concrete slab segment rested on Pier 1-S and Pier 2-S, each containing a mudline footing and

steel H-piles. The central pier in Case 3, Pier 3-S, contained two tapered rectangular pier

columns, with a 1.5 m (5 ft) wide impact face at approximately the same elevation as the top of a

small shear strut that spanned between the two 1.2 m (4 ft) thick waterline pile-cap segments.

The pier rested on eight battered 0.5 m (20 in) square prestressed concrete piles, each containing

a free length of approximately 3.7 m (12 ft).

The Case 3 FE model includes the southern channel pier and extends southward from the

centerline of barge traffic. The impacted pier, Pier 3-S, was constructed before the AASHTO

provisions were written (1991), and was flexible as it was not a channel pier. The pier was

48

located 115.8 m (380 ft) from the channel centerline, which was significantly closer to a distance

of three times the impacting vessel length, 138 m (450 ft), than the distance to the edge of the

navigation channel, 37.75 m (124 ft). Per the AASHTO specifications, the pier would be subject

to a reduced impact velocity, approaching that of the yearly mean current velocity

(Consolazio et al. 2002). The kinetic energy (Table 1-1) associated with an empty jumbo-hopper

barge drifting at the yearly mean current velocity for the Apalachicola Bay is representative of a

low-energy impact condition.

5.2.2 Case 4

Escambia Bay abuts Pensacola, Florida, in the southeastern United States. Case 4

(Fig. 5-1) consists of impact analysis of a model based on the Escambia Bay Bridge. Structural

components of this bridge model were derived from bridge plans developed in the 1960s. The

superstructure spanning from Pier 2-W to Pier 2-E consists of a 125 m (410 ft) continuous

three-span steel girder and concrete slab. A 28 m (92 ft) concrete girder-and-slab segment spans

the underlying concrete piers beyond Pier 2-E. All piers, except for the channel piers denoted as

Pier 1-E and Pier 1-W, contain two pier columns, a shear wall, pile cap, and waterline footing

foundation. The channel piers in Case 4 each contain two tapered rectangular pier columns, with

a 2.6 m (8.5 ft) wide head-on impact face at approximately the mid-height elevation of a 5.3 m

(17.5 ft) shear wall. The pier columns and shear wall overlie a 1.5 m (5 ft) thick mudline footing

and 1.8 m (6 ft) tremie seal. The channel pier foundations consist of eighteen battered and nine

plumb 0.6 m (24 in) square prestressed concrete piles.

The Case 4 FE model includes both of the channel piers and three auxiliary piers. The

impacted pier, Pier 1-E, was constructed before the AASHTO provisions were written (1991),

but contains large impact resistance relative to the impacted pier from Case 3, as Pier 1-E is a

channel pier. Impact on a channel pier with a relatively high impact resistance was chosen to

49

demonstrate the accuracy of the simplification algorithm for the medium-energy impact of a

fully-loaded jumbo-hopper barge and towboat, traveling at a higher speed than the mean

waterway velocity (Table 1-1).

5.2.3 Case 5

Case 5 (Fig. 5-2) consists of impact analysis of piers from the new St. George Island

Bridge, which replaced the old St. George Island Bridge in 2004. The structural model of the

new St. George Island Bridge was derived from construction drawings. Per these drawings,

Pier 46 through Pier 49 support five cantilever-constructed Florida Bulb-T girder-and-slab

segments at span lengths of 62.25 m (207.5 ft) for the channel and 78.5 m (257.5 ft) for the

flanking spans. Due to haunching, the depth of the post-tensioned girders vary from 2 m (6.5 ft)

at drop-in locations to 3.7 m (12 ft) at respective pier cap beam bearing locations. Simply

supported Florida Bulb-T beams with a depth equal to that of the haunched beams at the drop-in

locations span either side of Pier 50. All piers included in this model contain two pier columns, a

shear strut centered near a respective pier column mid-height, a pile cap, and a waterline footing

system. The central pier in Case 5, Pier 48, contains two round 1.8 m (6 ft) pier columns, a

(6.5 ft) thick pile cap, and fourteen battered and one plumb 1.4 m (4.5 ft) diameter prestressed

cylinder piles with a 3 m (10 ft) concrete plug extending earthward from the pile cap.

The new St. George Island Bridge was designed in accordance with current AASHTO

barge collision design standards and provided a means of validating the simplification algorithm

for barge impact energies similar to those used in present day design. The Case 5 FE model

includes both of the channel piers and three auxiliary piers. The impacted pier, Pier 48 was

designed for a static impact load of 14.48 MN (3255 kips). With respect to the static AASHTO

design impact load, an energy equivalent impact condition (Appendix D) is employed in Case 5.

The prescribed vessel mass and velocity yields an impact kinetic energy equivalent to four

50

fully-loaded jumbo class hopper barges and a towboat traveling slightly above typical waterway

vessel speeds for the Apalachicola Bay waterway (Table 1-1).

5.3 Comparison of Simplified and Full-Resolution Results

In bridge design applications related to waterway vessel collision, the analytically

quantified internal forces in a given pier structure govern subsequent structural component

sizing. Hence, accurate determination of internal forces is a necessary outcome of a bridge

structural analysis method. To highlight the ability of simplified analysis to accurately quantify

design forces over the full range of impacted pier structures, time-histories of internal shear force

induced by the impact loading are shown for the top of the impacted pier column and an

underlying pile-head node for Case 3 through Case 5 shown in Fig. 5-3 through Fig. 5-5,

respectively (additional comparisons of the impact force, displacements, and internal moments

are documented in Appendix C).

The predictions of load duration (the time during which the barge and pier are in contact),

common to both simplified and full-resolution analyses, are 0.26 sec, 0.78 sec, and 2.9 sec,

respectively, for Case 3, Case 4, and Case 5. At points in time greater than the respective load

durations, each bridge is in an unloaded condition and undergoes damped free-vibration.

Accordingly, pier response to time-history barge collision analysis may be divided into two

phases: first a load-phase then a free-vibration phase. In all three demonstration cases, peak

internal pier forces occur during the load-phase (0.13 sec, 0.17 sec, and 2.1 sec for Case 3,

Case 4, and Case 5, respectively). Therefore, agreement between the simplified and

full-resolution models is most critical during the load-phase, as forces obtained during this phase

ultimately govern bridge pier member design. Simplified analysis retains the ability to

accurately capture forces during the load-phase of response (Fig. 5-3 through Fig. 5-5 for each

case, respectively). Peak shear forces generated by full-resolution and simplified analysis during

51

the load-phase for each case differ by less than 2%. Reduced, yet still reasonable, agreement

with respect to period of response and subsequent peak values of shear force occur during the

free-phase of response for each case, however, such agreement is less critical and typically

irrelevant for design purposes.

Case 3 through Case 5 were analyzed on a Dell Latitude D610 notebook computer using

a single 2.13 GHz Intel PentiumM CPU and FB-MultiPier. The computation times necessary for

analysis completion of the simplified models were only 8%, 7.5%, and 8.4% of those required

for the full-resolution models of Case 3 through Case 5, respectively (Fig. 5-6). All cases

required significantly less than an hour to complete 800, 800, and 1600 time-steps of analysis,

respectively. Engineering judgment is required to determine the appropriate amount of analysis

time specified. However, analysis generally need not be conducted beyond the end of

load-phase, as evidenced by forces during the load-phase for Case 3 through Case 5.

5.4 Conclusions from Simplified-Coupled Analysis Demonstrations

Excellent agreement is observed during the load-phase response of the full-resolution and

simplified test cases, especially with respect to peak internal forces generated at various locations

of the impacted piers. From a design perspective, reasonable agreement between full and

simplified analytical results is also observed during the free-phase portions of respective

time-history responses. Time-histories of internal shear force, moment, and displacement are

adequately captured by the simplification algorithm, despite the simplifying stiffness and mass

assumptions that are made.

The time necessary to analyze the simplified models is significantly less than one hour in

each case, which is in contrast to the several hours necessary to analyze respective full-resolution

models. It should be noted that all FB-MultiPier analyses were conducted in compilation debug

52

mode. Considerable additional reduction in analysis time is expected if the same analyses were

to be conducted in a release compilation or commercial version of FB-MultiPier.

5.5 Dynamic Amplification of the Impacted Pier Column Internal Forces

Application of the simplification algorithm to each of the demonstration cases inherently

incorporates mass and acceleration based inertial forces that emerge from integration of the

dynamic system equations of motion. The simplification algorithm accurately captures dynamic

amplification of forces generated in the pier columns that would be absent from static analysis

results. Dynamic amplification in each case may be quantified by considering the maximum pier

column shears developed in models subjected to static application of the peak impact load

predicted through the coupled analysis. The peak shear and moments developed in the pier due

to static loading are then compared to those from the simplified and full-resolution dynamic

analyses (Fig. 5-7)

With respect to peak pier column structural demand, the dynamic analyses are in

excellent agreement with each other for all cases. However, the peak magnitudes of the

statically generated shears and moments, respectively, correspond to 59% and 64% of the

magnitude of the dynamically obtained counterparts for Case 3; and, 38% and 37%, respectively,

for Case 4 (Fig. 5-7). In each of these cases, a static analysis employing a dynamically obtained

peak impact load leads to un-conservative predictions of peak pier column demand, as static

analysis only encompasses stiffness considerations. In contrast, dynamic analyses incorporate

both stiffness and inertial effects associated with the superstructure and therefore capture

dynamic amplification of pier column forces due to the mass of the superstructure. Furthermore,

the simplified procedure retains the ability to capture pier column force amplification as

evidenced by the agreement between the simplified and full-resolution output pertaining to peak

pier column demand.

53

The impact energy specified in Case 5 is of sufficient magnitude to cause the barge and

impacted pier to remain in contact for a time greater than several periods of the fundamental pier

vibration mode. Consequently, the inertial forces in the impacted pier begin to dissipate due to

damping effects. This is evidenced by attenuation of oscillation exhibited in the pile head shear

force time-history for Case 5 from 0.1 sec to 2.5 sec (Fig. 5-5B). Despite the continued dynamic

activity in the top of the Pier 48 pier columns throughout the analysis (Fig. 5-5A), the overall

pier behavior approaches that of a static response as the impact load approaches a maximum

value. Additionally, because the AASHTO barge bow force-crush relationship (Fig. 4-1)

maintains a positive stiffness regardless of crush depth, the Case 5 peak impact force occurs at a

time in which the dynamic component of behavior of Pier 48 has substantially diminished.

Therefore, the peak pier column demands are driven by a static response in this case. As a result,

there is not a great difference between dynamic and static response (Fig. 5-7).

54

Pier 2-W Pier 1-W Pier 1-E Pier 2-E Pier 3-E

Impact


55

Pier 46 Pier 47 Pier 48 Pier 49 Pier 50

Impact


56

Time (s)

Forc

e (k

N)

Forc

e (k

ips)

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2-300

-200

-100

0

100

200

300

400

500

600

-50

-25

0

25

50

75

100

125

Simplified ModelFull-Resolution Model

A

Time (s)

Forc

e (k

N)

Forc

e (k

ips)

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2-100

-50

0

50

100

150

200

-15

0

15

30Simplified ModelFull-Resolution Model

B

Figure 5-3. Comparison of Case 3 simplified and full-resolution coupled analyses. A) Pier column top node horizontal shear. B) Pile head node horizontal shear.

57

Time (s)

Forc

e (k

N)

Forc

e (k

ips)

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2-300

-200

-100

0

100

200

300

400

500

600

700

-50

-25

0

25

50

75

100

125

150


A

Time (s)

Forc

e (k

N)

Forc

e (k

ips)

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2-50

0

50

100

150

200

250

0

15

30

45


B


58

Time (s)

Forc

e (k

N)

Forc

e (k

ips)

0 0.5 1 1.5 2 2.5 3 3.5 4-750

-500

-250

0

250

500

750

1000

1250

-120

-60

0

60

120

180

240


A

Time (s)

Forc

e (k

N)

Forc

e (k

ips)

0 0.5 1 1.5 2 2.5 3 3.5 4-100

0

100

200

300

400

500

600

700

800

0

25

50

75

100

125

150

175


B


59

Tim

e (m

in)

Tim

e (h

rs)

0

60

120

180

240

300

360

420

480

540

600

660

0

1

2

3

4

5

6

7

8

9

10

11

Case 3 Case4 Case 5

Simplified modelFull-resolution model

Figure 5-6. Time computation comparison of coupled analyses.

60

Case 3 Case 4 Case 5

0.53

7 M

N(1

21 k

ip)

0.53

9 M

N(1

21 k

ip)

0.31

6 M

N(7

1.0

kip)

0.69

1 M

N(1

55 k

ip)

0.69

9 M

N(1

57 k

ip)

0.25

9 M

N(5

8.2

kip)

10.8

MN

(243

0 ki

p)

10.6

MN

(239

0 ki

p)

10.6

MN

(239

0 ki

p)

Simplified dynamic analysis Full-resolution dynamic analysis Static analysis

A

Case 3 Case 4 Case 5

Simplified dynamic analysis Full-resolution dynamic analysis Static analysis

2.85

MN

-m(2

100

kip-

ft)

2.86

MN

-m(2

110

kip-

ft)

1.83

MN

-m(1

350

kip-

ft)

4.28

MN

-m(3

150

kip-

ft)

4.33

MN

-m(3

190

kip-

ft)

1.62

MN

-m(1

190

kip-

ft)

26.3

MN

-m(1

9400

kip

-ft)

25.1

MN

-m(1

8500

kip

-ft)

25.7

MN

-m(1

9000

kip

-ft)

B

Figure 5-7. Comparison of demonstration case simplified, full-resolution, and static analyses. A) Peak pier column shear. B) Peak pier column moment.

61

CHAPTER 6 CONCLUSIONS AND RECOMMENDATIONS FOR FUTURE RESEARCH

6.1 Conclusions

Numerical coupled analysis has been validated using experimental findings from the

2004 full-scale barge impact experiments. As one means of making coupled analysis feasible for

use in design settings, a simplified and standardized barge bow stiffness curve has been

recognized as desirable. Due to the scarcity of barge bow force-crush relationship data in the

literature, the AASHTO crush-curve has been selected. However, data specific to a particular

vessel type obtained by other means may easily be integrated into the coupled analysis

procedure. As an additional facilitation for the use of coupled analysis in design settings, an

algorithm has been presented that reduces a multi-span, multiple-pier model to a multi-span

single pier model with lateral and rotational springs, and lumped masses. With regard to the

stiffness approximation associated with the simplification algorithm, linear elastic lateral and

rotational springs have been shown to retain sufficient accuracy in respective simplified models

for design purposes despite the associated uncoupling of the respective DOF.

Three five-pier bridge analysis cases have been presented and subjected to the coupled

analysis procedure at simplified and full resolutions. Comparison of the results demonstrates the

ability of the simplification algorithm to predict time-history results in agreement with

full-resolution models for low, medium, and high-energy impact conditions through a range of

pier impact resistances. The simplified algorithm, used in conjunction with coupled analysis,

provides a feasible means of conducting barge-bridge collision analysis in design settings.

Required analyses times associated with simplified analysis are reduced to levels suitable for

design situations. Furthermore, the simplification algorithm retains analytical sophistication

62

sufficient to adequately quantify inertial bridge forces and the resulting distribution of internal

forces throughout a given pier.

Dynamic phenomena documented in previous barge-pier collision research, such as

dynamic amplification of pier column shear forces due to dynamic excitation of superstructure

elements, are quantified for three cases and compared to results obtained from a static analysis

procedure. Simplified coupled analysis is shown to adequately and efficiently capture such

effects and is found to be suitable for future incorporation into design provisions.

6.2 Recommendations for Future Research

Based on the advances made in this study, the following topics warrant additional future

investigation:

• The development of experimental procedures leading to a standardized body of crush-curves, including phenomena such as post-yield softening and unloading;

• High-resolution modeling or experimental testing of multiple-barge flotilla impacts, resulting in data sufficient to quantify any significant interactions between multiple barge flotillas; this would be in relation to improving the state-of-the-art SDF impact model; and,

• Possible revision of the AASHTO Probability of Collapse term.

63

APPENDIX A SUPPLEMENTARY COUPLED ANALYSIS VALIDATION DATA

The 2004 full-scale experiments (Consolazio et al. 2006) consisted of three distinct impact test setups, two of which are of interest in this study: the first impact tests were conducted on the stiff channel pier, Pier 1-S; the second set of tests were conducted on a flexible pier, Pier 3-S, with the superstructure intact for one span to the north and multiple spans to the south. After development of the barge force-crush relationship, coupled analyses were conducted on FB-MultiPier models of the Pier 1-S and Pier 3-S partial bridge structure at impact energies corresponding to the impact test events.

The highest impact energies, and therefore the most appreciable impact loads and

structure response, occurred during tests four through seven on Pier 1-S (termed test P1T4 through P1T7). Due to the flexibility of Pier 3-S, and the non-destructive nature of the testing, impact energies employed in the multi-span B3 bridge tests were considerably lower than that of the P1 test series. Even so, the second through fourth tests (termed test B3T2 through B3T4) generated considerable pier response and significant impact loads. Tests associated with significant loading or pier response were selected for validation of the coupled analysis procedure. Pertinent output from such analyses is included in this appendix. All P1 series analyses included here were conducted using the payload modifications discussed in Chapter 3.

64

0 0.25 0.5 0.75 1 1.25 1.50.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

Dis

plac

emen

t (in

)

Impact Point Displacement Time History

0 0.25 0.5 0.75 1 1.25 1.50

200

400

600

800

1000

1200

Time (sec)

Forc

e (k

ips)

Impact Force Time History

0 1 2 3 4 5 60

200

400

600

800

1000

1200

Analytical outputExperimental dataInput loading curve


Crush (in)

Forc

e (k

ips)

Barge Force Crush Output

Figure A-1. Analytical output comparison to experimental P1T4 barge impact data.

65

0 0.25 0.5 0.75 1 1.25 1.50.2

0.1

0

0.1

0.2

0.3

0.4

0.5

Time (sec)

Dis

plac

emen

t (in

)


0 0.25 0.5 0.75 1 1.25 1.50

200

400

600

800

1000

1200

Time (sec)

Forc

e (k

ips)


0 1 2 3 4 5 60

200

400

600

800

1000

1200



Crush (in)

Forc

e (k

ips)



66

0 0.25 0.5 0.75 1 1.25 1.50.5

0.25

0

0.25

0.5

0.75

Time (sec)

Dis

plac

emen

t (in

)


0 0.25 0.5 0.75 1 1.25 1.50

200

400

600

800

1000

1200

Time (sec)

Forc

e (k

ips)


0 1 2 3 4 5 6 7 8 90

200

400

600

800

1000

1200



Crush (in)

Forc

e (k

ips)



67

0 0.25 0.5 0.75 1 1.25 1.50.25

0

0.25

0.5

0.75

Time (sec)

Dis

plac

emen

t (in

)


0 0.25 0.5 0.75 1 1.25 1.50

200

400

600

800

1000

1200

Time (sec)

Forc

e (k

ips)


0 1 2 3 4 5 6 7 8 90

200

400

600

800

1000

1200



Crush (in)

Forc

e (k

ips)



68

0 0.25 0.5 0.75 1 1.25 1.50.5

0.25

0

0.25

0.5

0.75

1

Time (sec)

Dis

plac

emen

t (in

)


0 0.25 0.5 0.75 1 1.25 1.50

50

100

150

200

250

Time (sec)

Forc

e (k

ips)


0 0.5 1 1.5 2 2.5 30

50

100

150

200

250



Crush (in)

Forc

e (k

ips)


Figure A-5. Analytical output in comparison to experimental B3T2 barge impact data.

69

0 0.25 0.5 0.75 1 1.25 1.50.5

0.25

0

0.25

0.5

0.75

1

Time (sec)

Dis

plac

emen

t (in

)


0 0.25 0.5 0.75 1 1.25 1.50

50

100

150

200

250

Time (sec)

Forc

e (k

ips)


0 0.5 1 1.5 2 2.5 30

50

100

150

200

250



Crush (in)

Forc

e (k

ips)



70

0 0.25 0.5 0.75 1 1.25 1.50.750.5

0.250

0.250.5

0.751

1.251.5

1.75

Time (sec)

Dis

plac

emen

t (in

)


0 0.25 0.5 0.75 1 1.25 1.50

50100150200250300350400450

Time (sec)

Forc

e (k

ips)


0 0.5 1 1.5 2 2.5 3 3.50

50100150200250300350400450



Crush (in)

Forc

e (k

ips)



71

APPENDIX B CONDENSED UNCOUPLED STIFFNESS MATRIX CALCULATIONS

Within the discussion of the condensed uncoupled stiffness matrix, presented in Chapter 4, the condensed off-diagonal stiffness term that couples rotation and horizontal shear force ( ) is shown to produce relatively negligible shear forces with respect to the applied impact load. This affords the uncoupling of the condensed stiffness matrix of extraneous non-impacted portions of a given bridge model. This appendix contains comparisons of the same off-diagonal stiffness term ( ), alternatively viewed as a coupling between horizontal translation and a vertical moment, and the moment produced by the diagonal rotational stiffness term ( ) of the condensed stiffness matrix when the B3 numerical model (Fig.

couplingK

couplingK

θK3-3) is subject to an arbitrary static load at the impact location. A comparison of the

diagonal and off-diagonal moments reveals that the off-diagonal stiffness of piers adjacent to the impacted pier in a given full-resolution model may be neglected without sacrificing any appreciable analytical accuracy of forces developed in the impacted pier.

72

Off-Diagonal stiffness quantification: Case 3 Numerical Model

1. Obtain condensed stiffness matrix of Pier 2S to Pier 1S portion of full-resolution model

1.1 Apply unit lateral load at location of stiffness condensation; in this case, the center of the pier cap beam of Pier 2S

1.1.1 Store lateral translation and vertical rotation in appropriate entries of condensed flexibility matrix

1.2 Apply unit vertical moment at location of stiffness condensation; in this case, the center of the pier cap beam of Pier 2S

1.2.1 Store lateral translation and vertical rotation in appropriate entries of condensed flexibility matrix

Condensed flexibility matrix: FlexP2

0.00181781

8.5443− 10 7−⋅

12

8.5443− 10 7−⋅

12

5.28799 10 9−⋅

12

⎛⎜⎜⎜⎜⎝

⎞⎟⎟⎟⎟⎠

:= in/kip and rad/kip-in

The first row diagonal entry pertains to shear force per unit lateral translation; the second row diagonal pertains to vertical moment per unit vertical rotation

2. Invert condensed flexibility matrix to obtain condensed stiffness matrix

StiffP2 FlexP2 1−:=

Condensed stiffness matrix: StiffP2553.616

8.945 104×

8.945 104×

2.284 109×

⎛⎜⎜⎝

⎞⎟⎟⎠

= kip/in and kip-in/rad

73

Off-Diagonal stiffness quantification: Case 3 Numerical Model (Cont'd)

3. Apply peak static load at impact point of full-resolution model and record displacements

Static load applied at Node 109 of Pier 3-S:

PeakLoad 413.8:= kips

Induced displacements at location of condensed stiffness:

Vertical rotation:

θz 1.994 10 4−⋅:= rad

Horizontal translation:

Δx 0.1908:= in

4. Calculate moment due to diagonal stiffness term and vertical rotation

Mzdiagonal StiffP22 2, θz⋅:= Mzdiagonal 4.554 105×= kip-in

5. Calculate moment due to off-diagonal stiffness term and horizontal translation

Mzoffdiagonal StiffP21 2, Δx⋅:= Mzoffdiagonal 1.707 104×= kip-in

6. Compare magnitudes of "diagonal" and "off-diagonal" moments

ratioMzdiagonal

Mzoffdiagonal:= ratio 26.681=

The "diagonal" moment is significantly larger than the "off-diagonal" moment.

74

Flexibility Approximation: Case 3 Numerical Model7. Directly invert diagonal flexibility terms recorded in 1.2.1 of Off-Diagonal Stiffness Quantification

7.1 Approximation of Translational Stiffness Term

AppStiffTrans1

FlexP21 1,:= AppStiffTrans 550.112= kip/in

7.2 Approximation of Rotational Stiffness Term

AppStiffRot1

FlexP22 2,:= AppStiffRot 2.269 109

×= kip-rad/in

8. Calculate percent difference between approximated stiffness terms and stiffness terms obtained by flexibility matrix inversion (the latter terms being calculated in 2. of Off-Diagonal Stiffness Quantification)

8.1 Percent difference of translational stiffness term

AppStiffTrans StiffP21 1,−

StiffP21 1,100⋅ 0.633−= percent

8.2 Percent difference of rotational stiffness term

AppStiffRot StiffP22 2,−

StiffP22 2,100⋅ 0.633−= percent

The approximation yields nominally different values of stiffness.

75

APPENDIX C SIMPLIFIED-COUPLED ANALYSIS CASE OUTPUT

To further bolster the assertion that the simplification algorithm predicts impacted pier response with an accuracy that, within reason, matches that of full-resolution bridge coupled analysis, additional time-history data from each of Case 3 through Case 5 are included in this appendix. More specifically, time-histories of shear, moment, and displacement are provided at the pier column top and pile head for each case. Additionally, barge force-crush data obtained from simplified and full-bridge analyses are included. Accompanying this data are the impact location displacement time-history and impact location force time-history for each of Case 3 through Case 5. Consequently, the data presented in Fig. C-1 through Fig. C-9 were obtained using the AASHTO barge bow force-crush relationship. Finally, data obtained from the same pier models are presented when simplified and full-resolution analyses are conducted using a bilinear barge bow force-crush relationship with an initial stiffness and “shift point” (see Chapter 4) identical to that found in the AASHTO curve. Output pertaining to these analyses are located in Fig. C-10 through Fig. C-18.

2

0 0.25 0.5 0.75 1 1.25 1.5 1.75 21

0.5

0

0.5

1

1.5

2

Time (sec)

Dis

plac

emen

t (in

)


0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

100

200

300

400

500

600

Time (sec)

Forc

e (k

ips)


0 1 2 3 4 50

300

600

900

1200

1500

Two-span single-pierFive-pierInput loading curve


Crush (in)

Forc

e (k

ips)


Figure C-1. Case 3 AASHTO curve coupled analysis output comparison at impact location.

77

Shear Force Time History

Moment Time History

0 0.25 0.5 0.75 1 1.25 1.5 1.75 21

0.5

0

0.5

1

1.5

Two-span single-pierFive-pierTwo-span single-pierFive-pier

Time (sec)

Dis

plac

emen

t (in

)

Displacement Time History

0 0.25 0.5 0.75 1 1.25 1.5 1.75 22000

1500

1000

500

0

500

1000

1500

Time (sec)

Mom

ent (

kip-

ft)

0 0.25 0.5 0.75 1 1.25 1.5 1.75 275

50

25

0

25

50

75

100

125

Time (sec)

Forc

e (k

ips)

Figure C-2. Case 3 AASHTO curve coupled analysis output comparison at pier column top.

78

0 0.25 0.5 0.75 1 1.25 1.5 1.75 220

10

0

10

20

30

40

Time (sec)

Forc

e (k

ips)


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2500

400

300

200

100

0

100

200

Time (sec)

Mom

ent (

kip-

ft)

Moment Time History

0 0.25 0.5 0.75 1 1.25 1.5 1.75 21

0.5

0

0.5

1

1.5

2


Time (sec)

Dis

plac

emen

t (in

)


Figure C-3. Case 3 AASHTO curve coupled analysis output comparison at pile head.

79

0 0.25 0.5 0.75 1 1.25 1.5 1.75 20.25

0

0.25

0.5

0.75

1

1.25

1.5

Time (sec)

Dis

plac

emen

t (in

)


0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

200

400

600

800

1000

1200

1400

1600

Time (sec)

Forc

e (k

ips)


0 2 4 6 8 10 12 14 160

200

400

600

800

1000

1200

1400

1600



Crush (in)

Forc

e (k

ips)



80

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2755025

0255075

100125150175

Time (sec)

Forc

e (k

ips)


0 0.25 0.5 0.75 1 1.25 1.5 1.75 22500

2000

1500

1000

500

0

500

1000

Time (sec)

Mom

ent (

kip-

ft)

Moment Time History

0 0.25 0.5 0.75 1 1.25 1.5 1.75 20.5

0

0.5

1

1.5

2

2.5


Time (sec)

Dis

plac

emen

t (in

)



81

0 0.25 0.5 0.75 1 1.25 1.5 1.75 210

0

10

20

30

40

50

60

Time (sec)

Forc

e (k

ips)


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2600

500

400

300

200

100

0

100

Time (sec)

Mom

ent (

kip-

ft)

Moment Time History

0 0.25 0.5 0.75 1 1.25 1.5 1.75 20.25

0

0.25

0.5

0.75

1

1.25


Time (sec)

Dis

plac

emen

t (in

)



82

0 25 50 75 100 125 150 175 2000

500

1000

1500

2000

2500

3000

3500



Crush (in)

Forc

e (k

ips)

0 0.5 1 1.5 2 2.5 3 3.5 40

500

1000

1500

2000

2500

3000

3500

Time (sec)

Forc

e (k

ips)

0 0.5 1 1.5 2 2.5 3 3.5 40.25

0

0.25

0.5

0.75

1

1.25

Time (sec)

Dis

plac

emen

t (in

)





83

0 0.5 1 1.5 2 2.5 3 3.5 4150

100

50

0

50

100

150

200

250

Time (sec)

Forc

e (k

ips)


0 0.5 1 1.5 2 2.5 3 3.5 42500200015001000500

0500

10001500

Time (sec)

Mom

ent (

kip-

ft)

Moment Time History

0 0.5 1 1.5 2 2.5 3 3.5 40.5

0.25

0

0.25

0.5

0.75

1

1.25


Time (sec)

Dis

plac

emen

t (in

)



84

0 0.5 1 1.5 2 2.5 3 3.5 4200

150

100

50

0

50

Time (sec)

Forc

e (k

ips)


0 0.5 1 1.5 2 2.5 3 3.5 41000

750

500

250

0

250

Time (sec)

Mom

ent (

kip-

ft)

Moment Time History

0 0.5 1 1.5 2 2.5 3 3.5 40.25

0

0.25

0.5

0.75

1


Time (sec)

Dis

plac

emen

t (in

)



85

2

0 0.25 0.5 0.75 1 1.25 1.5 1.75 21

0.5

0

0.5

1

1.5

2

Time (sec)

Dis

plac

emen

t (in

)


0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

100

200

300

400

500

600

Time (sec)

Forc

e (k

ips)


0 1 2 3 4 50

300

600

900

1200

1500



Crush (in)

Forc

e (k

ips)


Figure C-10. Case 3 bilinear curve coupled analysis output comparison at impact location.

86


Moment Time History

0 0.25 0.5 0.75 1 1.25 1.5 1.75 21

0.5

0

0.5

1

1.5


Time (sec)

Dis

plac

emen

t (in

)


0 0.25 0.5 0.75 1 1.25 1.5 1.75 22000

1500

1000

500

0

500

1000

1500

Time (sec)

Mom

ent (

kip-

ft)

0 0.25 0.5 0.75 1 1.25 1.5 1.75 275

50

25

0

25

50

75

100

125

Time (sec)

Forc

e (k

ips)

Figure C-11. Case 3 bilinear curve coupled analysis output comparison at pier column top.

87

0 0.25 0.5 0.75 1 1.25 1.5 1.75 220

10

0

10

20

30

40

Time (sec)

Forc

e (k

ips)


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2500

400

300

200

100

0

100

200

Time (sec)

Mom

ent (

kip-

ft)

Moment Time History

0 0.25 0.5 0.75 1 1.25 1.5 1.75 21

0.5

0

0.5

1

1.5

2


Time (sec)

Dis

plac

emen

t (in

)


Figure C-12. Case 3 bilinear curve coupled analysis output comparison at pile head.

88

0 0.25 0.5 0.75 1 1.25 1.5 1.75 20.25

0

0.25

0.5

0.75

1

1.25

1.5

Time (sec)

Dis

plac

emen

t (in

)


0 0.25 0.5 0.75 1 1.25 1.5 1.75 20

200

400

600

800

1000

1200

1400

1600

Time (sec)

Forc

e (k

ips)


0 2 4 6 8 10 12 14 160

200

400

600

800

1000

1200

1400

1600



Crush (in)

Forc

e (k

ips)



89

0 0.25 0.5 0.75 1 1.25 1.5 1.75 2100

50

0

50

100

150

Time (sec)

Forc

e (k

ips)


0 0.25 0.5 0.75 1 1.25 1.5 1.75 22500200015001000500

0500

10001500

Time (sec)

Mom

ent (

kip-

ft)

Moment Time History

0 0.25 0.5 0.75 1 1.25 1.5 1.75 21

0.5

0

0.5

1

1.5

2

2.5


Time (sec)

Dis

plac

emen

t (in

)



90

0 0.25 0.5 0.75 1 1.25 1.5 1.75 210

0

10

20

30

40

50

60

Time (sec)

Forc

e (k

ips)


0 0.25 0.5 0.75 1 1.25 1.5 1.75 2600

500

400

300

200

100

0

100

Time (sec)

Mom

ent (

kip-

ft)

Moment Time History

0 0.25 0.5 0.75 1 1.25 1.5 1.75 20.25

0

0.25

0.5

0.75

1

1.25


Time (sec)

Dis

plac

emen

t (in

)



91

0 30 60 90 120 150 180 210 240 270 3000

250

500

750

1000

1250

1500



Crush (in)

Forc

e (k

ips)

0 1 2 3 4 5 60

250

500

750

1000

1250

1500

Time (sec)

Forc

e (k

ips)

0 1 2 3 4 5 60.25

0

0.25

0.5

0.75

Time (sec)

Dis

plac

emen

t (in

)





92

0 1 2 3 4 5 6150

100

50

0

50

100

150

200

250

Time (sec)

Forc

e (k

ips)


0 1 2 3 4 5 62500

2000

1500

1000

500

0

500

1000

Time (sec)

Mom

ent (

kip-

ft)

Moment Time History

0 1 2 3 4 5 60.25

0

0.25

0.5

0.75

1


Time (sec)

Dis

plac

emen

t (in

)



93

0 1 2 3 4 5 6125

100

75

50

25

0

25

Time (sec)

Forc

e (k

ips)


0 1 2 3 4 5 6600

500

400

300

200

100

0

100

Time (sec)

Mom

ent (

kip-

ft)

Moment Time History

0 1 2 3 4 5 60.1

0

0.1

0.2

0.3

0.4

0.5

0.6


Time (sec)

Dis

plac

emen

t (in

)



94

APPENDIX D ENERGY EQUIVALENT AASHTO IMPACT CALCULATIONS

The new St. George Island Bridge was designed and constructed after the AASHTO vessel collision specifications went in effect, hence, the piers of this bridge were designed to resist barge impact loading. Furthermore, pier impact load data was contained within the bridge plans used to develop the numerical model for Case 5 of this thesis. The initial kinetic energy specified in the Case 5 coupled analyses was derived from the known design impact load and pertinent equations found in AASHTO. Conversely, energy-equivalent static AASHTO impact loads are calculated from the kinetic energies employed in Case 3 and Case 4. The barge width modification factor is 1.0 in the following calculations as a jumbo-hopper is selected as the impacting vessel in all demonstration cases.

95

Back calculation of AASHTO impact force using Case 3 Impact Energy

The Old St. George Island Bridge Pier 3-S was subject to an impact energy consisting of:

Barge weight

W 1.78 MN⋅:= W 200.08T= W 181.478 tonne=

Barge velocity

V 1.03ms

:= V 2.002knot= V 3.379fts

=

Assume a hydrodynamic mass coefficient of Ch 1.05:=

Impact energy

KE12

Ch W⋅g

⋅ V2⋅:= KE 74.564kip ft⋅= KE 0.101MN m⋅=

From the American Association of State and Highway Transportation Officials (AASHTO) GuideSpecification and Commentary for Vessel Collision Design of Highway Bridges , the equations forbarge crush depth and kinetic energy associated with impact are:

Barge crush depth:

abKE

56721+⎛⎜

⎝⎞⎟⎠

0.51−

⎡⎢⎣

⎤⎥⎦

10.2⋅:= ab 0.067= ft

The energy equivalent AASHTO static impact force is:

Pb Pb 1349 110 ab⋅+( )← ab 0.34≥if

Pb 4112ab← ab 0.34<if

Pbreturn

:=

Pb 274.787= kip

96

Back calculation of AASHTO impact force using Case 4 Impact Energy

The Escambia Bay Bridge Pier 1-E was subject to an impact energy consisting of:

Barge weight

W 18 MN⋅:= W 2.023 103× T= W 1.835 103

× tonne=

Barge velocity

V 1.54ms

:= V 2.994knot= V 5.052fts

=

Assume a hydrodynamic mass coefficient of Ch 1.05:=

Impact energy

KE12

Ch W⋅g

⋅ V2⋅:= KE 1.686 103

× kip ft⋅= KE 2.285MN m⋅=

From the American Association of State and Highway Transportation Officials (AASHTO) GuideSpecification and Commentary for Vessel Collision Design of Highway Bridges , the equations forbarge crush depth and kinetic energy, and barge width associated with impact are:

Barge crush depth:

abKE

56721+⎛⎜

⎝⎞⎟⎠

0.51−

⎡⎢⎣

⎤⎥⎦

10.2⋅:= ab 1.417= ft

The energy equivalent AASHTO static impact force is:

Pb Pb 1349 110 ab⋅+← ab 0.34≥if

Pb 4112ab← ab 0.34<if

Pbreturn

:=

Pb 1.505 103×= kip

97

Back calculation of Case 5 impact energy using AASHTO impact force

From bridge plans of the New St. George Island Bridge Channel Pier, the design impact load is:

Pb 3255:= kips

From the American Association of State and Highway Transportation Officials (AASHTO) GuideSpecification and Commentary for Vessel Collision Design of Highway Bridges , the equations forbarge crush depth and kinetic energy associated with impact are:

Barge crush depth:

ab abPb 1349−

110←

Pb 1349−110

0.34≥if

abPb

4112←

Pb4112

0.34<if

abreturn

:=

ab 17.327= ft

Kinetic energy associated with impact:

KEab

10.21+⎛⎜

⎝⎞⎟⎠

21−

⎡⎢⎣

⎤⎥⎦

5672⋅:= KE 3.564 104×= kip-ft

Define flotilla design velocity as a function of Hydrodynamic Mass Coefficient and Flotilla weight (tonnes)

V CH W,( ) KE 29.2⋅CH W⋅

⎛⎜⎝

⎞⎟⎠

0.5:=

Assume Hydrodynamic Mass Coefficient is 1.05.

Define weight of barge as a function of the number of barges in the flotilla; assume towboat weighs 120 tons (US, short)

W n( )n 1700 200+( )⋅ 120+

1.102311311:=

98

Back calculation of Case 5 impact energy using AASHTO impact force(Cont'd)

Try using four fully loaded Jumbo Hopper barges and check that the accompanying velocity is attainable within the waterway.

The weight of four fully loaded Jumbo Hopper barges and the tow boat is:

W 4( ) 7.003 103×= tonnes

The velocity of the flotilla, necessary to generate a static impact load of 3255 kips is:

V 1.05 W 4( ),( ) 11.896= ft/sec

Conclusion: the four barge flotilla is a reasonable number of barges for use in a single column flotilla in the southeastern United States, and 11.896 ft/sec is an attainable speed in the St. George Island waterway as typical traveling speeds are: 10.13 ft/sec (Consolazio et al. 2002).

99

REFERENCES

AASHTO. (1991). Guide Specification and Commentary for Vessel Collision Design of Highway Bridges, American Association of State Highway and Transportation Officials, Washington, D.C.

Arroyo, J. R., Ebeling, R. M., and Barker, B. C. (2003). “Analysis of Impact Loads from Full-Scale Low-Velocity, Controlled Barge Impact Experiments, December 1998.” US Army Corps of Engineers Report ERDC/ITL TR-03-3, 2003.

Consolazio, G. R., Cook, R. A., and Lehr, G. B. (2002). “Barge Impact Testing of the St. George Island Causeway Bridge Phase I : Feasibility Study.” Structures Research Report No. 783, Engineering and Industrial Experiment Station. University of Florida, Gainesville, Florida, January.

Consolazio, G. R. and Cowan, D. R. (2003). “Nonlinear Analysis of Barge Crush Behavior and its Relationship to Impact Resistant Bridge Design.” Computers and Structures, Vol. 81, Nos.8-11, pp. 547-557.

Consolazio, G. R., Lehr, G. B., and McVay, M. C. (2004a). “Dynamic Finite Element Analysis of Vessel-Pier-Soil Interaction During Barge Impact Events.” Transportation Research Record: Journal of the Transportation Research Board. No. 1849, Washington, D.C., pp. 81-90.

Consolazio, G. R., Hendrix, J. L., McVay, M. C., Williams, M. E., and Bollman, H. T. (2004b). “Prediction of Pier Response to Barge Impacts Using Design-Oriented Dynamic Finite Element Analysis.” Transportation Research Record: Journal of the Transportation Research Board. No. 1868, Washington, D.C., pp. 177-189.

Consolazio, G. R. and Cowan, D. R. (2005). “Numerically Efficient Dynamic Analysis of Barge Collisions with Bridge Piers.” ASCE Journal of Structural Engineering, ASCE, Vol. 131, No. 8, pp. 1256-1266.

Consolazio, G. R., Cook, R. A., and McVay, M. C. (2006). “Barge Impact Testing of the St. George Island Causeway Bridge”, Structures Research Report No. 2006/26868, Engineering and Industrial Experiment Station, University of Florida, Gainesville, Florida, March.

FB-MULTIPIER User’s Manual. (2007). Florida Bridge Software Institute, University of Florida, Gainesville, Florida.

FB-PIER User’s Manual. (2003). Florida Bridge Software Institute, University of Florida, Gainesville, Florida.

Goble, G., Schulz, J., and Commander, B. (1990). Lock and Dam #26 Field Test Report for The Army Corps of Engineers, Bridge Diagnostics Inc., Boulder, CO.

100

Hendrix, J. L. (2003). “Dynamic Analysis Techniques for Quantifying Bridge Pier Response to Barge Impact Loads.” Masters Thesis, Department of Civil and Coastal Engineering, Univ. of Florida, Gainesville, Fla.

Larsen, O. D. (1993). “Ship Collision with Bridges: The Interaction between Vessel Traffic and Bridge Structures.” IABSE Structural Engineering Document 4, IABSE

Knott, M., and Prucz, Z. (2000). Vessel Collision Design of Bridges: Bridge Engineering Handbook, CRC Press LLC.

Meier-Dörnberg, K. E. (1983). “Ship Collisions, Safety Zones, and Loading Assumptions for Structures in Inland Waterways.” Verein Deutscher Ingenieure (Association of German Engineers) Report No. 496, 1983, pp. 1-9.

McVay, M. C., Wasman, S. J., Bullock, P. J. (2005). St. George Geotechnical Investigation of Vessel Pier Impact, Engineering and Industrial Experiment Station, University of Florida, Gainesville, Florida.

Patev, R. C., Barker, B. C., and Koestler, L. V., III. (2003). “Full-Scale Barge Impact Experiments, Robert C. Byrd Lock and Dam, Gallipolis Ferry, West Virginia.” United States Army Corps of Engineers Report ERDC/ITL TR-03-7, December.

Yuan, P. (2005). “Modeling, Simulation and Analysis of Multi-Barge Flotillas Impacting Bridge Piers.” PhD dissertation, Dept. of Civil Engineering, Univ. of Kentucky, Lexington, Ky.

101

BIOGRAPHICAL SKETCH

Michael Davidson was born in Louisville, Kentucky. He enrolled at the University of

Kentucky in August 2000. After being awarded the National Science Foundation Graduate

Research Fellowship and obtaining his Bachelor of Science in civil engineering from the

University of Kentucky (summa cum laude) in May 2005, he began graduate school at the

University of Florida in the College of Engineering, Department of Civil and Coastal

Engineering. The author will receive his Master of Science degree in August 2007, with a

concentration in structural engineering. Upon graduation, the author will continue his education

at the University of Florida, ultimately earning a degree of Doctor of Philosophy with a

specialization in structural engineering.

102

Documents

By MICHAEL THOMAS DAVIDSON - University of Floridaufdcimages.uflib.ufl.edu/UF/E0/02/14/72/00001/davidson_m.pdf · SIMPLIFIED DYNAMIC BARGE COLLISION ANALYSIS FOR BRIDGE PIER DESIGN