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Information and Technology Laboratory erdc/itltr- 16 -i

US Army Corps
of Engineers®

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Development Center

INNOVATIVE SOLUTIONS
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Navigation Systems Research Program

Simplified Dynamic Structural Time-History
Response Analysis of Flexible Approach Walls
Founded on Clustered Pile Groups Using
Impact.Deck

Barry C. White, Jose Ramon Arroyo, and Robert M. Ebeling July 2016

Cross-

Section

Barge A
Impact

Cross-Section

McAlpine Alternative
Impact Flexible Wall

— Q 0 - <g r( p - Q ( p —

Barge

Impact

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Navigation Systems Research Program

ERDC/ITL TR-16-1
July 2016

Simplified Dynamic Structural Time-History
Response Analysis of Flexible Approach Walls
Founded on Clustered Pile Groups Using
Impact.Deck

Barry C. White and Robert M. Ebeling

Information Technology Laboratory

U.S. Army Engineer Research and Development Center

3909 Halls Ferry Road

Vicksburg, MS 39180-6199

Jose Ramon Arroyo

Department of General Engineering
University of Puerto Rico
Mayaguez, PR 00681

Final report

Approved for public release; distribution is unlimited.

Prepared for U.S. Army Corps of Engineers
441 G. Street NW
Washington, DC 20314-1000

under Work Unit Number 448769

ERDC/ITL TR-16-1

Abstract

Flexible approach walls are being considered for retrofits, replacements, or
upgrades to Corps lock structures that have exceeded their economic
lifetime. This report discusses a new engineering software tool to be used
in the design or evaluation of flexible approach walls founded on clustered
pile groups and subjected to barge train impact events.

This software tool, Impact_Deck, is used to perform a dynamic, time-
domain analysis of three different types of pile-founded flexible approach
walls: an impact deck, an alternative flexible approach wall, and a guard
wall. Dynamic loading is performed using Impact_Force time histories
(Ebeling et al. 2010). This report covers the numerical methods used to
create this tool, a discussion of the graphical user interface for the tool,
and an analysis of results for the three wall systems.

The results of analyzing the three wall systems reveals that dynamic
evaluations should be performed for these structures because of inertial
effects occurring in the wall superstructure and substructure. These
inertial effects can cause overall and individual response forces that are
greater than the peak force from the Impact_Force time history.

This report also discusses the advantages of load sharing between multiple
pile groups in an approach wall substructure. In the case of Lock and Dam
3, the peak reaction force for any individual pile group was 11% of the peak
impact load.

DISCLAIMER: The contents of this report are not to be used for advertising, publication, or promotional purposes.
Citation of trade names does not constitute an official endorsement or approval of the use of such commercial products.
All product names and trademarks cited are the property of their respective owners. The findings of this report are not to
be construed as an official Department of the Army position unless so designated by other authorized documents.

DESTROY THIS REPORT WHEN NO LONGER NEEDED. DO NOT RETURN IT TO THE ORIGINATOR.

ERDC/ITL TR-16-1 iii

Contents

Abstract.ii

Figures and Tables.vi

Preface.xiii

Unit Conversion Factors.xv

1 Dynamic Structural Time-History Response Analysis of a Flexible Approach Wall

Supported by Clustered Pile Groups during Impact with a Barge Train.1

1.1 Introduction - Glancing Impact Blows and Flexible Approach Wall Structural

Systems.1

1.2 Examples of the Next Generation Flexible Approach Walls.1

1.2.1 Lock and Dam 3 .1

1.2.2 McAlpine flexible wall .3

1.2.3 Guard walls .4

1.3 Overview of Dynamic Time-Flistory Response Analysis of a Flexible Beam

Supported Over Elastic-Plastic Spring Supports.4

1.4 Report Contents.8

2 Impact Deck Approach Wall - Lock and Dam 3 Example.11

2.1 Introduction.11

2.2 Lock and Dam 3 - Physical Model.11

2.3 Lock and Dam 3 - Construction Drawings.18

2.4 Lock and Dam 3 - Mathematical Model.18

2.5 Nonlinear force-deflection relationship for the spring supports.31

2.6 Solving for the motion of the structure.32

2.7 Validation of lmpact_Deck Computer Program.32

2.8 Numerical Example of the Elastic-Plastic Response Using lmpact_Deck.34

2.9 lmpact_Deck GUI results.36

2.10 Final Remarks.56

3 An Approach Wall with Impact Beams on Nontraditional Pile Supported Bents -

McAlpine Example.57

3.1 Introduction.57

3.2 Alternative Flexible Approach Wall - Physical Model.57

3.3 McAlpine Alternative Flexible Approach Wall - Mathematical Model.58

3.4 Nonlinear force-deflection relationship for the springs supports.60

3.5 Solving for the motion of the structure.62

3.6 Validation of lmpact_Deck Computer Program.63

3.7 Numerical Example of the Elastic-Plastic Response Using lmpact_Deck.66

3.8 lmpact_Deck GUI results.67

3.9 Final Remarks.84

4 Traditional Impact Beam Guard Walls.85

ERDC/ITL TR-16-1 iv

4.1 Introduction.85

4.2 Guard Walls - Physical Model.85

4.3 Guard wall - Mathematical Model.86

4.4 Nonlinear force-deflection relationship for the springs supports.88

4.5 Solving for the motion of the structure.90

4.6 Validation of lmpact_Deck Computer Program.90

4.7 Numerical Example of the Elastic-Plastic Nonlinear Response Using

lmpact_Deck.92

4.8 lmpact_Deck GUI results.93

4.9 Final Remarks.110

5 lmpact_Deck Graphical User Interface (GUI).Ill

5.1 Introduction.Ill

5.2 Geometry Tab.112

5.3 Impact Time History Tab.118

5.4 Beam Properties Tab.119

5.5 Pile Cluster Spring Tab.121

5.6 Analyze Tab.123

5.7 Output Tab.126

5. 7.1 FEO Nodal Output . 127

5.7.2 FEO Element Output . 133

5.7.3 FEO Pile Group Response . 138

5.7.4 Run Information . 143

5.8 Example: Geometry Input for the Impact Deck at Lock and Dam 3.144

5.9 Final Remarks.148

6 Conclusions and Recommendations.149

References.153

Appendix A: Lock and Dam 3 - Equations of Motion for the Mathematical Model.155

Appendix B: McAlpine Alternative Flexible Wall - Equations of Motion for the

Mathematical Model.165

Appendix C: Guard wall - Equations of Motion for the Mathematical Model.179

Appendix D Push-over analysis for batter-pile bent system.189

Appendix E: Formulation for the rotational spring stiffness for the McAlpine flexible

approach wall clustered group of vertical piles model.224

Appendix F: HHT-a method.232

Appendix G: Wilson-6 Method.235

Appendix H: Member End Release Details for Load Applied at the End Release Node.238

Appendix I: Rayleigh Damping

240

ERDC/ITL TR-16-1

Appendix J: Key lmpact_Deck Program Variables.245

Report Documentation Page

ERDC/ITL TR-16-1 vi

Figures and Tables

Figures

Figure 1.1 Front-end cells of Lock and Dam 3.2

Figure 1.2 Lock and Dam 3 cross-section and plan view.3

Figure 1.3 McAlpine flexible walls.4

Figure 1.4 Guard wall schematic drawing.5

Figure 1.5 Barge train impacting at a fixed impact position along the simply supported,
flexible impact beam mathematical model with the barge train oriented at an approach
angle 0 to the wall’s XGiobai axis (plan view).6

Figure 1.6 Example of an impact pulse-force time history.6

Figure 1.7 Barge impact point force moving along the wall from initial contact time ti to

final contact time t 2 ..7

Figure 2.1 Front end cell of Lock and Dam 3.12

Figure 2.2 Pinned connection between the circular cell and the Impact Deck at Lock and

Dam 3.13

Figure 2.3 Arrangement of piles at Lock and Dam 3.13

Figure 2.4 Impact Deck supported over piles at Lock and Dam 3.14

Figure 2.5 An upstream view starting at the concrete cell of the pile-founded flexible

approach wall at Lock and Dam 3.14

Figure 2.6 Precast bases before installation.15

Figure 2.7 Precast bases during installation.15

Figure 2.8 Precast bases connected to piles.16

Figure 2.9 Construction joint between concrete block segments (axial, shear and moment
transfer connection).16

Figure 2.10 Construction joints.17

Figure 2.11 Massive concrete circular cell.17

Figure 2.12 Pile-founded flexible impact deck structure.18

Figure 2.13 Lock and Dam 3 guide wall plan view 1 of 2.19

Figure 2.14 Lock and Dam 3 guide wall plan view 2 of 2.20

Figure 2.15 Lock and Dam 3 guide wall detail of pile layout and end cell.21

Figure 2.16 Lock and Dam 3 guide wall section view.22

Figure 2.17 Lock and Dam 3 guide wall plan - 5.23

Figure 2.18 Lock and Dam 3 guide wall plan - 6.24

Figure 2.19 Lock and Dam 3 flexible approach wall.25

Figure 2.20 (a) Typical 3-D segment of the impact-deck beam element, (b) Impact force
applied to the Impact Deck, (c) Typical 3-D beam element.25

Figure 2.21 Typical 2-D beam element used in lmpact_Deck.26

Figure 2.22 Lock and Dam 3 mathematical model.27

Figure 2.23 First beam element used in Lock and Dam 3 mathematical model.27

Figure 2.24 Last beam element used in Lock and Dam 3 mathematical model.27

ERDC/ITL TR-16-1 vii

Figure 2.25 Description of two beam elements connected at the inter-monolith

connection.28

Figure 2.26 Beam elements numbering in a typical internal monolith.28

Figure 2.27 Cross-section of Lock and Dam 3.29

Figure 2.28 Transverse and longitudinal push-over results for a single row of three piles
aligned in the transverse direction.30

Figure 2.29 Force-displacement relation of the spring support.32

Figure 2.30 Force time history of Winfield Test # 10.33

Figure 2.31 Validation of lmpact_Deck against SAP2000.34

Figure 2.32 Dynamic response of the transverse spring located at x = 402.896 ft. .35

Figure 2.33 Dynamic response of the transverse spring located at x = 402.896 ft. .36

Figure 2.34 Reprint of the Figure 4-4 transverse direction of loading push-over analyses

from Ebeling et al. (2012); fixed head results in green and pinned head results in blue.37

Figure 2.35 lmpact_Deck GUI Table of maximum nodal displacements for the L&D3

example impact deck.39

Figure 2.36 Transverse nodal displacement time histories for node 86.40

Figure 2.37 Longitudinal nodal displacement time histories for node 137.40

Figure 2.38 Rotational nodal displacement time histories for node 93.41

Figure 2.39 Transverse wall displacements at 0.26 sec.42

Figure 2.40 Longitudinal wall displacements at 0.21 sec.42

Figure 2.41 Rotational wall displacements at 0.26 sec.43

Figure 2.42 lmpact_Deck GUI table of element minimum and maximum axial forces for

the L&D3 example impact deck.43

Figure 2.43 lmpact_Deck GUI table of element minimum and maximum shear forces for

the L&D3 example impact deck.44

Figure 2.44 lmpact_Deck GUI table of element minimum and maximum moments for the
L&D3 example impact deck.44

Figure 2.45 Axial-force time histories for element 82.45

Figure 2.46 Axial-force time histories for element 81.46

Figure 2.47 Shear-force time histories for element 82.46

Figure 2.48 Shear-force time histories for element 81.47

Figure 2.49 Moment time histories for element 81.47

Figure 2.50 Moment time histories for element 82.48

Figure 2.51 Wall axial forces at 0.2 sec.48

Figure 2.52 Wall shear forces at 0.2 sec.49

Figure 2.53 Wall moments at 0.18 sec.49

Figure 2.54 Table of pile group response maximum forces and moments and their time.50

Figure 2.55 Table of pile group response maximum displacements and their time.51

Figure 2.56 Table of pile group responses for each pile group individually and summed

(not shown) at time 0.26 sec.51

Figure 2.57 Transverse pile group responses for the pile group at node 85 and at time

0.24 sec.52

ERDC/ITL TR-16-1 viii

Figure 2.58 Longitudinal pile group responses for the pile group at node 85 and at time

0.24 sec.52

Figure 2.59 Rotational pile group response for the pile group at node 85 and at time 0.24 sec.53

Figure 2.60 Time-history plot of transverse input forces, total force response for all the

pile groups, and an individual pile group response forces.54

Figure 3.1 McAlpine alternative flexible approach wall.58

Figure 3.2 McAlpine flexible approach wall mathematical model.59

Figure 3.3 (a) Typical 3-D segment of the Impact Deck beam element, (b) Impact force

applied to the Impact Deck, (c) Typical 3-D beam element.60

Figure 3.4 Typical 2-D beam element used in lmpact_Deck.60

Figure 3.5 Plan view of the flexible wall pile layout.61

Table 3.1 Primary loading curve for the transverse spring model for a McAlpine alternative
flexible wall bent (3 piles).62

Figure 3.6 Force time history of Winfield Test # 10.64

Figure 3.7 Validation of lmpact_Deck against SAP2000 - Transverse displacement at

node 1.64

Figure 3.8 Validation of lmpact_Deck against SAP2000 - Transverse displacement at

node 23.65

Figure 3.9 Validation of lmpact_Deck against SAP2000 - Transverse displacement at

node 12’.65

Figure 3.10 Validation of lmpact_Deck against SAP2000 - Rotation at node 12 and 12’.66

Figure 3.11 Dynamic transverse response of node 12 and 12’.68

Figure 3.12 Dynamic response of the rotational spring at node 12 and 12’.68

Figure 3.13 Dynamic response of the transverse spring located at x = 84.5 ft. .69

Figure 3.14 lmpact_Deck GUI table of maximum nodal displacements for the McAlpine

flexible wall.70

Figure 3.15 Transverse nodal displacement time histories for node 21.70

Figure 3.16 Longitudinal nodal displacement time histories for node 22.71

Figure 3.17 Rotational nodal displacement time histories for node 22.71

Figure 3.18 Transverse wall displacements at 0.252 sec.72

Figure 3.19 Longitudinal wall displacements at 0.192 sec.72

Figure 3.20 Rotational wall displacements at 0.22 sec.73

Figure 3.21 lmpact_Deck GUI table of element minimum and maximum axial forces for

the McAlpine flexible wall example.74

Figure 3.22 Axial-force time histories for element 21.75

Figure 3.23 Axial-force time histories for element 20.75

Figure 3.24 Shear-force time histories for element 21.76

Figure 3.25 Shear-force time histories for element 20.76

Figure 3.26 Moment time histories for element 31.77

Figure 3.27 Moment time histories for element 21.77

Figure 3.28 Wall axial forces at 0.2 sec.78

Figure 3.29 Wall shear forces at 0.220 sec.78

Figure 3.30 Wall moments at 0.220 sec.79

ERDC/ITL TR-16-1 ix

Figure 3.31 Table of pile group response maximum displacements.79

Figure 3.32 Response forces for the pile groups at time 0.2200 sec.80

Figure 3.33 Response forces for the pile groups at time 0.2800 sec.80

Figure 3.34 Pile group response for the pile group at node 2 and at time 0.298 sec.81

Figure 3.35 Time-history plot of transverse input forces, total force response for all the

pile groups, and an individual pile group response forces.82

Figure 4.1 Guard wall schematic drawing.86

Figure 4.2 Guard flexible approach wall mathematical model.87

Figure 4.3 (a) Typical 3-D segment of the Impact Deck beam element, (b) Impact force

applied to the Impact Deck, (c) Typical 3-D beam element.87

Figure 4.4 Typical 2-D beam element used in lmpact_Deck.88

Figure 4.5 Force-displacement relations from push-over analysis of a single guard wall pile.89

Figure 4.6 Force time history of Winfield Test # 10.91

Figure 4.7 Validation of lmpact_Deck against SAP2000 - Transverse displacement at

node 1.91

Figure 4.8 Validation of lmpact_Deck against SAP2000 - Transverse displacement at

node 6.92

Figure 4.9 Dynamic transverse response of node 1.93

Figure 4.10 Dynamic transverse response of spring at node 6.94

Figure 4.11 Plastic force-displacement of the transverse spring at node 6.94

Figure 4.12 lmpact_Deck GUI pile group longitudinal and transverse spring model

backbone curves.95

Figure 4.13 lmpact_Deck GUI table of maximum nodal displacements for the guard wall.96

Figure 4.14 Transverse nodal displacement time histories for node 51.97

Figure 4.15 Longitudinal nodal displacement time histories for node 51.97

Figure 4.16 Rotational nodal displacement time histories for node 50.98

Figure 4.17 Transverse wall displacements at 0.332 sec.98

Figure 4.18 Longitudinal wall displacements at 0.41 sec.99

Figure 4.19 Rotational wall displacements at 0.286 sec.99

Figure 4.20 lmpact_Deck GUI table of element minimum and maximum axial forces for

the guard wall Example.100

Figure 4.21 Axial force time histories for element 51.101

Figure 4.22 Axial force time histories for element 50.101

Figure 4.23 Shear force time histories for element 51.102

Figure 4.24 Shear force time histories for element 50.102

Figure 4.25 Moment time histories for element 51.103

Figure 4.26 Moment time histories for element 50.103

Figure 4.27 Wall axial forces at 0.410 sec.104

Figure 4.28 Wall shear forces at 0.714 sec.104

Figure 4.29 Wall moments at 0.716 sec.105

Figure 4.30 Table of pile group response maximum displacements.105

Figure 4.31 Forces at the three pile group nodes at time 0.332 sec.106

ERDC/ITL TR-16-1

Figure 4.32 Transverse pile group response for the pile group at node 51 and at time

0.332 sec.106

Figure 4.33 Time-history plot of transverse input forces, total force response for all the

pile groups, and an individual pile group response forces.108

Figure 5.1 Introducing lmpact_Deck.Ill

Figure 5.2 Geometry for a flexible wall.112

Figure 5.3 Geometry for a guard wall.113

Figure 5.4 Geometry for an lmpact_Deck.113

Figure 5.5 Zooming in the input plot section.116

Figure 5.6 The zoomed view.116

Figure 5.7 Selected nodes are highlighted.116

Figure 5.8 Entering an offset to copy selected nodes.117

Figure 5.9 Confirming the offset copy (which can be performed multiple times).117

Figure 5.10 Selected nodes are copied at the offset position.118

Figure 5.11 Input for an impact time history.119

Figure 5.12 Extending a time history with 0.0 value.119

Figure 5.13 Beam properties tab as it appears for a flexible wall.120

Figure 5.14 Beam properties tab as it appears for an impact deck or guard wall.120

Figure 5.15 Beam properties tab as it appears for a flexible wall.121

Figure 5.16 Beam properties tab as it appears for an lmpact_Deck.122

Figure 5.17 Analyze tab input for analysis method and specified output.124

Figure 5.18 Selecting finite element nodes where data will be captured.125

Figure 5.19 Output tab for selecting and viewing select data.126

Figure 5.20 Output tab with selected data.127

Figure 5.21 FEO nodal output window showing maximum nodal values for all the nodes.128

Figure 5.22 Graph of node 70 X-displacement vs time.129

Figure 5.23 Graph of node 70 Y-displacement vs time.129

Figure 5.24 Graph of node 70 Z-displacement vs time.130

Figure 5.25 Animated graph of wall X-displacement.132

Figure 5.26 Animated graph of wall Y-displacement.132

Figure 5.27 Animated graph of wall Z-displacement.133

Figure 5.28 FEO element output window showing maximum and minimum force and
moments acting on all the elements.134

Figure 5.29 Graph showing element 60 axial force vs time.135

Figure 5.30 Graph showing element 60 moment force-length vs time.135

Figure 5.31 Graph showing element 60 shear force vs time.136

Figure 5.32 Animated graph of axial forces acting on the wall.137

Figure 5.33 Animated graph of moments acting on the wall.137

Figure 5.34 Animated graph of shears acting on the wall.138

Figure 5.35 Pile Group Response Maximum and Minimum Forces and Moments.139

Figure 5.36 Pile Group Response Peak Deflections.140

ERDC/ITL TR-16-1 xi

Figure 5.37 Pile Group Response Maximum and Minimum Forces and Moments.140

Figure 5.38 Animated plot of node 85 X-force and displacement vs time.141

Figure 5.39 Animated plot of node 85 Y-force and displacement vs time.142

Figure 5.40 Animated plot of node 85 Z-force and displacement vs time.142

Figure 5.41 Output run information with selected FEO node output.144

Figure 5.42 Add node at position 0.0 ft as an inter-monolith node.145

Figure 5.43 Add interpolated nodes from 3.3541666 ft to 101.4791666 ft.146

Figure 5.44 Selecting nodes with a left-mouse, click-drag.146

Figure 5.45 Selected nodes are shown with vertical lines.146

Figure 5.46 Selected nodes can be copied multiple times with the copy selected nodes

button.147

Figure 5.47 The copy selected nodes dialog lets the user specify an offset.147

Figure 5.48 Select OK the number of times that the selected nodes need to be copied.147

Figure 5.49 Finally, Add the Final Node.148

Figure A.l Shape function for axial displacement effect.157

Figure A.2 Shape function for transverse displacement and rotation effect.159

Figure A.3 Force-displacement relation of the spring support.164

Figure B.l Shape function for axial displacement effect.167

Figure B.2 Shape function for transverse displacement and rotation effect.169

Figure B.3 Force-displacement relation of the spring support.174

Figure B.4 Typical McAlpine flexible wall system.175

Figure B.5 McAlpine flexible wall mathematical model.175

Figure B.6 Transformation of beam element coordinate system (Local-Global).178

Figure C.l Shape function for axial displacement effect.181

Figure C.2 Shape function for transverse displacement and rotation effect.183

Figure C.3 Force-displacement relation of the spring support.188

Figure D.l Pipe pile approach wall.189

Figure D.2 CPGA analytical model.190

Figure D.3 Simple interaction diagram for 24-inch-diameter pipe pile.191

Figure D.4 (After Figure 3 Yang 1966) Coefficient of critical buckling strength.193

Figure D.5 (After Figure 9 Yang 1966) Coefficient decrement of buckling strength.193

Figure D.6, (After Figure 7 Yang 1966) Coefficient of horizontal load capacity.197

Figure D.7 (After Figure 2 Yang 1966) Effective embedment of pile at buckling.202

Figure D.8 Load - displacement plot for pipe pile system.222

Figure E.l. Plan view of the McAlpine flexible alternative approach wall system.224

Figure E.2 Relation between Global-Axis and central support Local-Axis.225

Figure E.3 Location of the Center of Rigidity.225

Figure E.4 Definition of the forces and distances generated when the pile cap rotate.228

Figure E.5 Rotational angle definition when the pile cap rotate.229

Figure G.l Linear variation of acceleration over normal and extended time steps.235

Figure 1.1 (a) Mass-proportional damping; (b) stiffness-proportional damping.240

ERDC/ITL TR-16-1 xii

Tables

Table 2.1 Primary loading curve for the transverse spring model of a single pile group with
a leading vertical pile followed by two batter piles (Lock and Dam 3).38

Table 2.2 Primary loading curve for the longitudinal spring model of a single pile group

with a leading vertical pile followed by two batter piles (Lock and Dam 3).38

Table 2.3 Maximum nodal displacements for the lmpact_Deck example problem.39

Table 2.4 Extreme forces/moments for the lmpact_Deck example problem.45

Table 2.5 Transverse forces with respect to time.54

Table 3.1 Primary loading curve for the transverse spring model for a McAlpine alternative
flexible wall bent (3 piles).62

Table 3.2 Primary loading curve for the longitudinal spring model for a McAlpine

alternative flexible wall bent (3 piles).62

Table 3.3 Primary loading curve for a spring model for a single 6-ft diameter DIP pile.62

Table 3.4 Maximum nodal displacements for the McAlpine flexible wall example problem.70

Table 3.5 Extreme forces/moments for the lmpact_Deck example problem.74

Table 3.6 Transverse forces with respect to time.82

Table 4.1 Primary loading curve for the transverse spring model for a bent with two

vertical piles.89

Table 4.2 Primary loading curve for the longitudinal spring model for a bent with two

vertical piles.90

Table 4.3 Maximum nodal displacements for the guard wall example problem.96

Table 4.4 Extreme Forces/Moments for the Impact Deck Example Problem.100

Table 4.5 Transverse forces with respect to time.107

Table D.l Euler critical buckling load - translating pile top - pinned head condition.195

Table D.2 Euler critical buckling load - translating pile top - fixed head condition.196

Table D.3 Euler critical buckling load - translating pile top - pinned head condition.209

Table D.4 Euler critical buckling load - translating pile top - fixed head condition.210

Table F.l. HHT-a Method.234

Table G.l Wilson’s Method: Nonlinear Systems.237

ERDC/ITL TR-16-1

xiii

Preface

More than 50% of the U.S. Army Corps of Engineers’ locks and their
approach walls have continued past their economic lifetimes. As these
structures wear out, they must be retrofitted, replaced, or upgraded with a
lock extension. Innovative designs must be considered for the Corps
hydraulic structures, particularly flexible approach walls, and new tools for
evaluating these designs must be developed.

This technical report describes an engineering methodology for the
dynamic structural response analysis of a flexible approach wall consisting
of a series of impact beams or impact decks supported by clustered pile
groups during barge train impact loading. This engineering methodology
is implemented in a PC-based FORTRAN program and Visual Modeler
named Impact_Deck, which is also discussed in this report. The
engineering formulation for Impact_Deck uses an impact-force time
history acting on the clustered pile group founded flexible impact beams
or impact decks to characterize the impact event. This impact-force time
history may be obtained using a companion program Impact_Force
(Ebeling et al. 2010).

This report was authorized by Headquarters, U.S. Army Corps of
Engineers (HQUSACE), and was written from October 2013 to March
2014. It was published under the Navigation Systems Research Program,
Work Unit “Flexible Approach Walls.” Jeff McKee was the HQUSACE
Navigation Business Line Manager.

The Program Manager for the Navigation Systems Research Program was
Charles Wiggins, Coastal and Hydraulics Laboratory (CHL), U.S. Army
Engineer Research and Development Center (ERDC). Jeff Lillycrop was
Technical Director, CHL-ERDC. The research was led by Dr. Robert M.
Ebeling, Information Technology Laboratory (ITL), ERDC, under the
general supervision of Dr. Reed L. Mosher, Director, ITL-ERDC; Patti S.
Duett, Deputy Director, ITL-ERDC. This work effort was also done under
the general supervision of Dr. Robert M. Wallace, Chief, Computational
Science and Engineering Division (CSED), ITL, during the initial
formulation and programming stage. During the data interpretation and
report writing stages, Elias Arredondo, Dr. Kevin Abraham, and
Dr. Jerrell R. Ballard were Acting Division Chiefs. Dr. Ballard is the CSED

ERDC/ITL TR-16-1

xiv

Chief for the final stage of the publication process. Dr. Ebeling was the
Principal Investigator of the “Flexible Approach Walls” work unit.

This report was written by Barry C. White of ITL-ERDC, Professor Jose
Ramon Arroyo, University of Puerto Rico at Mayaguez, and Dr. Ebeling of
ITL-ERDC. White is with the Computational Analysis Branch (CAB), of
which Elias Arredondo is Chief.

Impact_Deck software formulation was developed by Arroyo and Ebeling.
Input specifications for the pertinent engineering features and boundary
conditions of the three different wall types to be analyzed by this software
were provided by Ebeling, White, and Arroyo. Programming for the
engineering formulation was led by Arroyo, with support from White and
Ebeling. The Graphical User Interface (GUI) comprised of the Visual
Modeler and an extensive and detailed Output Visualization software
package was developed by White with support from Arroyo and Ebeling.
Example problem input definition was provided by Ebeling, Arroyo, and
White. Initial engineering program validation and example problem
engineering assessments were led by Arroyo. Initial engineering
formulation documentation was created by Arroyo, with support by
Ebeling and White. Example problem output was collected and interpreted
by White and Arroyo. White was lead on the final organization of this
report, including the compilation, recasting, and reduction of the
engineering formulation description and example problems using the
Impact_Deck processor, with the aid of Arroyo. Having provided the
Visual Pre- and Post-Processor for Impact_Deck, White led in writing the
user interface and output visualization sections for the report. White also
provided additional example problems for verification and to support
observations highlighting the unique engineering advantages of each of
the three flexible approach wall systems, with the aid of the Visual Pre-
and Post-Processor. These unique features were interpreted from the data
results by Ebeling and White.

At the time of publication, COL Bryan S. Green was Commander, ERDC,
and Dr. Jeffery P. Holland was the Director.

ERDC/ITL TR-16-1

Unit Conversion Factors

Multiply

To Obtain

cubic feet

0.02831685

cubic meters

cubic inches

1.6387064 E-05

cubic meters

cubic yards

0.7645549

cubic meters

degrees (angle)

0.01745329

radians

feet

0.3048

meters

foot-pounds force

1.355818

joules

inches

0.0254

meters

inch-pounds (force)

0.1129848

newton meters

knots

0.5144444

meters per second

microns

1.0 E-06

meters

miles (nautical)

1,852

meters

miles (U.S. statute)

1,609.347

meters

miles per hour

0.44704

meters per second

pounds (force)

4.448222

newtons

pounds (force) per foot

14.59390

newtons per meter

pounds (force) per inch

175.1268

newtons per meter

pounds (force) per square foot

47.88026

pascals

pounds (force) per square inch

6.894757

kilopascals

pounds (mass)

0.45359237

kilograms

pounds (mass) per cubic foot

16.01846

kilograms per cubic meter

pounds (mass) per cubic inch

2.757990 E+04

kilograms per cubic meter

pounds (mass) per square foot

4.882428

kilograms per square meter

pounds (mass) per square yard

0.542492

kilograms per square meter

slugs

14.59390

kilograms

square feet

0.09290304

square meters

square inches

6.4516 E-04

square meters

tons (force)

8,896.443

newtons

tons (force) per square foot

95.76052

kilopascals

tons (2,000 pounds, mass)

907.1847

kilograms

tons (2,000 pounds, mass) per square foot

9,764.856

kilograms per square meter

yards

0.9144

meters

ERDC/ITL TR-16-1

1 Dynamic Structural Time-History

Response Analysis of a Flexible Approach
Wall Supported by Clustered Pile Groups
during Impact with a Barge Train

1.1 Introduction - Glancing Impact Blows and Flexible Approach
Wall Structural Systems

A glancing blow impact event of a barge train impacting an approach wall as
it aligns itself with a lock is an event of short duration. The contact time
between the impact corner of the barge train and the approach wall can
range from one second to several seconds. In order to reduce construction
costs as well as to reduce damage to barges during glancing blow impacts
with lock approach walls, the next generation of Corps approach walls are
more flexible than the massive, stiff-to-rigid structures constructed in the
past. A flexible approach wall or flexible approach wall system is one in
which the wall/system has the capacity to absorb impact energy by
deflecting or “flexing” during impact, thereby affecting the dynamic impact
forces developing during the impact event. Pile-founded approach wall
structural systems are characterized as flexible structures. This report
summarizes an engineering methodology as well as the corresponding
software for performing a dynamic structural response analysis. The
analysis is of a flexible impact approach wall deck or impact beam system
supported by piles representing using an elastic-plastic spring model of
each of the clustered pile groups to a barge train impact event. The PC-
based software used for conducting the dynamic structural response
analysis is referred to as Impact_Deck. A pulse-force time history of the
barge train impacting the flexible approach wall is used to dynamically load
the model. The PC-based software Impact_Force (Ebeling et al. 2010) is
used to develop the pulse-force time history required by the Impact_Deck
software.

1.2 Examples of the Next Generation Flexible Approach Walls

1.2.1 Lock and Dam 3

Lock and Dam No. 3 (Lock and Dam 3) is a lock and dam located near Red
Wing, Minnesota on the Upper Mississippi River around river mile 796.9.

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It was constructed and placed in operation July 1938. The site underwent
major rehabilitation from 1988 through 1991. In recent years, a guide wall
extension was added to the project.

The structure consists of eight reinforced concrete monoliths of 104 ft 10
in. each. It was constructed by joining eight concrete blocks of 12 ft 6 in.
with a free end of 37.5 in. at both ends. Each block is supported over two
rows of piles with three piles per row. That results in a monolith supported
by 48 piles, where 36 of these piles are battered with a batter of 1:4. Over
the piles, the monolith’s dimensions are 5 ft in height and 22 ft in width.
One end of the deck is pinned and connected to a massive circular concrete
cell and the other end is free. The inter-monolith connections are
connected by using four reinforcing steel bars just to transfer the axial and
shear force and no flexural moment transfer. The piles have a diameter of
2 ft, and are symmetrically located at each concrete block. The weight of
each precast concrete beam is approximately 6,919 tons. Some of the
drawings/plans of this flexible impact deck are shown in Appendix D.
Examples of this type of Corps flexible approach wall is shown in
Figures 1.1 and 1.2 for Lock and Dam 3.

Figure 1.1 Front-end cells of Lock and Dam 3.

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Figure 1.2 Lock and Dam 3 cross-section and plan view.

1.2.2 McAlpine flexible wall

McAlpine Locks and Dam are located in downtown Louisville, Kentucky.
The dam is at mile 604.4 of the Ohio River and the locks are in the
Louisville and Portland Canal on the Kentucky side of the river. The 56 ft x
600 ft auxiliary lock was completed in 1921. The no ft x 1200 ft main
chamber opened in 1961. A new lock chamber (110 ft x 1200 ft) began
operation in 2009.

The alternative flexible approach wall structure discussed in this section
consists of a continuous elastic concrete beam with segments spanning
approximately 96 ft in length. The continuity of the beam is achieved by
means of shear key at each pile group support. That means the beams just
transfer the longitudinal and transverse forces with no moment transfer at
each pile support. The axial and transverse forces at the end of the beams
are transferred to the pile cap by means of a shear key. The shear key has a
length of 11.5 ft. The length of the shear key is the distance between two
consecutive concrete beams. The shear key is part of the massive pile cap
that rests over the pile group. The pile group consists of three piles, each
one with a diameter of 5 ft 8 in. They are arranged in a triangular scheme
to absorb the torsion generated at the pile group due to the eccentricity
between the center of the pile group and the location of the end of the
beams that rest over the pile group. A plan view drawing and a cross-
section view are presented in Figure 1.3.

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Figure 1.3 McAlpine flexible walls.

1.2.3 Guard walls

This kind of flexible approach wall can be found at numerous Corps locks.
The structure consists of a continuous elastic concrete beam with a span of
approximately 50 or 60 ft long, each segment. The continuity of the beam
is achieved by means of shear key at each pile-group support. That means
the beams just transfer the longitudinal and transverse forces with no
moment transfer at each pile support. The axial and transverse forces at
the ends of the beams are transferred to the pile cap by means of a shear
key. The shear key is a concrete block between the end and start of two
consecutive flexible beams constraining the motion of the beam to the
motion of the pile bent. The length of the shear key is equal to the width of
the pile cap of the pile group. The shear key is part of the massive pile cap
that rests over the pile group. The pile group consists of two aligned piles,
each with a diameter of 5 ft 8 in. The two piles are arranged in such a way
that no torsion is transferred to the pile group. A plan view drawing and a
cross-section view are presented in Figure 1.4.

1.3 Overview of Dynamic Time-History Response Analysis of a

Flexible Beam Supported Over Elastic-Plastic Spring Supports

Due to the flexibility and the mass of the new generation flexible approach
walls, the impact event can be a dynamic event from the point of view of the
mathematical structural model. In structural dynamics the mathematical

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model of bodies of finite dimensions undergoing translatory motion are
governed by Newton’s Second Law of Motion, expressed as

^F = m»a (1.1)

where F are forces, m is mass, and a is acceleration at each time step t
during motion.

Figure 1.4 Guard wall schematic drawing.

In the mathematical model, the forces acting on the flexible wall mass at
each time step t are (l) the impact force at time step t, (2) the elastic
restoring forces (of the beam), and (3) the damping forces (of the beam).
This report discusses an engineering methodology that uses Equation 1.1
to compute the dynamic structural response of a flexible impact beam
supported over flexible supports of the mathematical model to the impact-
force time history. The impact event is idealized as shown in Figure 1.5 for
the mathematical beam and impact event model of the force time history,
Fnormal-wa u(t) is developed by scaling of existing pulse-force time histories
recorded during the full-scale barge impact experiments conducted at
Winfield Lock & Dam (Ebeling et al. 2010) and the Pittsburgh Prototype
tests (Patev et al. 2003). The force time history shown in Figure 1.6
denotes the component of the pulse force time history acting normal to the
wall. Initial barge train contact with the wall starts at time ti and ends at
time £2. The solution to this dynamic problem will be a succession of
solutions at user-specified time steps starting at time ti. These solution
time steps are dictated by the time step the user selects for the barge train
impact-force time history created using the PC-based software

ERDC/ITL TR-16-1

Impact_Force (Ebeling et al. 2010). Due to the nature of dynamic
structural response of some types of beams with consideration of both the
duration and frequency characteristics of the impact-force time history,
the peak response of the simply supported, flexible impact beam may
occur during impact (i.e., between times ti and t 2 ) or after impact
concluded (i.e., after time t 2 ). The engineering methodology discussed in
this report and corresponding software are capable of addressing both
types of dynamic structural systems responses.

Figure 1.5 Barge train impacting at a fixed impact position
along the simply supported, flexible impact beam mathematical
model with the barge train oriented at an approach angle 0 to
the wall’s Xe/obai axis (plan view).

Figure 1.6 Example of an impact pulse-force time history.

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An alternative formulation incorporated within the PC-based program
Impact_Deck allows for the specification of a barge train having an initial
contact with the impact beam at a position designated X_Impact (in PC-
program Impact_Deck input terminology) at time ti and moving in contact
at a constant velocity (V) along the beam as shown in Figure 1.7. The
position of the impact point force moves with time after contact. The time
after contact is designated as an increment in time Ati, and it occurs at an
absolute time of [ti + Ati]. The change in position of the contact force is
designated a distance AX from initial contact point X_Impact and is given
by

AX'= 7 • At,.

( 1 . 2 )

Figure 1.7 Barge impact point force moving along the wall from initial contact time ti to final contact time fa.

Flexible

The position of the point load along the beam at time ti is

[Xpoint load] t; =^- Im P aCt +A* (1.3)

Substituting from Equation 1.2, Equation 1.3 becomes

ERDC/ITL TR-16-1

[Xpoint loadlt, =[X_Impact +V.At ( ] (1.4)

Thus, the normal force time history of Figure 6 moves along the beam
from time ti to time t 2 according to the user-specified velocity ( V).

1.4 Report Contents

The engineering methodology discussed in this report is implemented in a
PC-based FORTRAN program named Impact_Deck, which is also discussed
in this report. A pulse-force time history (normal to the flexible approach
wall) is used in this dynamic time-history response analysis to represent the
demand made of the flexible beam supported over elastic-plastic springs
during the impact event. The impact-force time history to be used in the
Impact_Deck analysis is created by the companion PC-based program
Impact_Force (Ebeling et al. 2010). The engineering formulation for
Impact_Force uses the impulse momentum principle to convert the linear
momentum of a barge train into a pulse-force time history acting normal to
the approach wall.

The engineering formulation developed for and implemented in
Impact_Deck assumes that the District engineer will have knowledge of

1. Length, modulus of elasticity, cross-sectional area, moment of inertia and
mass per unit length (equal to the weight per unit length divided by the
gravitational constant, including hydrodynamic added mass for the beam)
of the flexible impact beam

2. Point of initial impact

3. Velocity (V) the barge train moves along the approach wall during impact

4. Dynamic coefficient of friction between the wall and the barge train

5. Force-displacement relationship of a pile group

6. Impact pulse-force time history normal to the impact beam

The engineering formulation developed for Impact_Force assumes that
the District engineer will have knowledge of

1. Size, the weight (and mass) of the barge train (including hydrodynamic
added mass)

2. Barge approach velocity (often expressed in local barge coordinates)

3. Approach angle (the angle measured from the face of the wall to the side of
the barge train)

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This information will be required for the usual, unusual, and extreme
design load cases.

Sections 2-4 describe the relationships that comprise the engineering
formulation used to solve Newton’s Second Law of Motion for the dynamic
response of a flexible impact beam supported on groups of piles and
subjected to a barge train impact. The groups of piles are modeled as stiff
elastic-plastic springs. The barge train impact is modeled using a
representative pulse-force time history applied normal to the point of
contact between the barge train and the flexible approach wall. In this
initial engineering formulation implemented in Impact_Deck, the
numerical solution of Newton’s Second Law of Motion for the impact
pulse-force time history is applied to the flexible impact beam. The
numerical solution makes use of the Wilson’s 0 method to solve the
equations of motion of the multiple degrees of freedom (MDOF) system in
the time domain. Each section discusses the results of this analysis for a
different structural system. The analysis in section 2 is for the Lock and
Dam 3 guide wall structural system; section 3 is for the McAlpine flexible
wall structural system; and section 4 is for a typical guard wall structural
system.

Section 5 introduces the user to the Graphical User Interface (GUI),
Engineering Processor, and Visual Post-Processor named Impact_Deck.
An example problem presents the features for input and the output
visualization.

Section 6 presents the conclusions of this report based on the
Impact_Deck computer software.

Appendix A discusses the formulation for a structural impact deck founded
on rows of pile groups. This type of flexible approach wall structural system
was used for the Lock and Dam 3 approach wall extension. Each pile group
cluster consists of an in-line row of vertical and batter piles.

Appendix B discusses the formulation for a flexible approach wall founded
on a triangular formation of three vertical piles modeled using
Impact_Deck. The McAlpine lock alternative flexible approach wall
structural system is an example of this type of flexible-impact structure.

Appendix C discusses the formulation for a simply supported flexible-
impact beam supported by two groups of clustered vertical piles modeled

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using Impact_Deck. Guard walls are an example of this type of flexible-
impact structure. Each pile-group cluster consists of an in-line row of
vertical piles.

Appendix D discusses the formulation for a two-translational spring model
of a row of piles that is used in Impact_Deck. This appendix discusses the
non-linear, force-deflection relationship for primary loading and for
unload-reload response of the clustered group of piles using the push-over
method of analysis and CPGA 1 software applied to the Lock and Dam 3
approach wall extension problem.

Appendix E discusses the formulation for a single rotational spring model
of a clustered group of vertical piles. This type of flexible approach wall
structural system, proposed for use at McAlpine Locks and Dam, is
referred to as the “alternative” flexible approach wall system.

Appendix F discusses the numerical method formulation for the time
integration of the Equation of Motion by HHT-a.

Appendix G discusses the numerical method formulation for the time
integration of the Equation of Motion by the Wilson -0 method.

Appendix H summarizes the member end release details used in
Impact_Deck for a load applied at the end release node.

Appendix I summarizes the Raleigh Damping formulation for the three
flexible approach walls.

Appendix J summarizes the Impact_Deck ASCII input file.

Appendix K lists key Impact_Deck FORTRAN program variables.

1 CPGA is CASE software for the Pile Group Analysis.

ERDC/ITL TR-16-1

2 Impact Deck Approach Wall - Lock and
Dam 3 Example

2.1 Introduction

This chapter summarizes an engineering methodology using the
Impact_Deck software for performing a dynamic structural response
analysis of a flexible impact deck supported over groups of clustered piles
and subjected to a barge train impact event. This example is based on the
real-world example at Lock and Dam 3 on the Mississippi River in
Minnesota.

This example models a glancing blow impact event of a barge train
impacting an approach wall as it aligns itself with a lock is an event of short
duration; the contact time between the impact corner of the barge train and
the approach wall can be as short as a second or as long as several seconds.
The next generation of Corps approach walls is more flexible than the
massive, stiff-to-rigid structures constructed in the past in order to reduce
construction costs as well as to reduce damage to barges during glancing
blow impacts with lock approach walls. A flexible approach wall or flexible
approach wall system is one in which the wall has the capacity to absorb
impact energy by deflecting or “flexing” during impact, thereby affecting the
dynamic impact forces that develop during the impact event.

2.2 Lock and Dam 3 - Physical Model

Lock and Dam 3 is located in Welch Township, Minnesota, on the Upper
Mississippi River, approximately 6 miles upstream from Red Wing,
Minnesota (around river mile 796).

Artist renderings, idealized cut-away sections, and pictures of Lock and
Dam 3 are shown in Figures 2.1 through 2.12, as well as in Figures D.i and
D.2 (Appendix D). The impact deck of the flexible guide wall consists of
eight reinforced concrete monoliths. Each monolith is approximately
104 ft 10 in. in length 1 . Each impact-deck monolith was constructed by
joining together eight concrete precast block segments that are 12 ft 6 in.
in length. Each 12 ft 6 in. block segment is supported over two rows of

1 The cited dimensions and those shown on construction drawings may deviate slightly from as-built
conditions.

ERDC/ITL TR-16-1

piles, with three piles per row. The front row of piles is vertical and the
back two rows of piles have a batter of 1:4. The piles are concrete-filled
pipe piles having a diameter of 2 ft and a vertical height of approximately
72 ft (Figure D.i). Figure 2.6 shows one of the pre-cast 12 ft 6 in. bases for
each block segment prior to installation. The base is hung from the top of
the piles (Figure 2.8). The impact-deck monolith is then cast on top of
eight neighboring block segments. Figure 2.2 shows the relationship of the
two and a half precast bases (the areas above the piles but below the
monolith, separated by vertical lines) to the overtopping monolith and
supporting pile groups. That results in an impact-deck monolith being
supported by 48 flexible piles, where 16 of these piles are vertical and
another 32 of these piles are battered. Over the piles, the impact-deck
monolith has dimensions of 5 ft height and 22 ft width.

Each end of adjoining monoliths is structurally detailed to provide for
shear and axial load transfer but no moment transfer between monoliths
using shear bars that are close to the line through the cross-section center.
The ends of each monolith are 37.5 in. from the last supporting pile group.

One end of the flexible guide wall impact deck (consisting of eight
monoliths) is pin-connected to a massive circular concrete cell and the other
end, adjacent to the existing approach wall, is free. The weight of impact
deck (all eight monoliths) is approximately 627,655 kg (6,919 tons).

Figure 2.1 Front end cell of Lock and Dam 3.

ERDC/ITL TR-16-1

Figure 2.2 Pinned connection between the circular cell and the Impact Deck at Lock and Dam 3.

Figure 2.3 Arrangement of piles at Lock and Dam 3.

ERDC/ITL TR-16-1

Figure 2.4 Impact Deck supported over piles at Lock and Dam 3.

Figure 2.5 An upstream view starting at the concrete
cell of the pile-founded flexible approach wall at Lock
and Dam 3.

ERDC/ITL TR-16-1

Figure 2.6 Precast bases before installation.

Figure 2.7 Precast bases during installation.

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Figure 2.8 Precast bases connected to piles.

Figure 2.9 Construction joint between concrete block segments (axial, shear
and moment transfer connection).

ERDC/ITL TR-16-1

Figure 2.10 Construction joints.

Figure 2.11 Massive concrete circular cell.

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Figure 2.12 Pile-founded flexible impact deck
structure.

2.3 Lock and Dam 3 - Construction Drawings

In Figures 2.13-2.18, some general construction drawings are provided
that demonstrate the pile arrangement, the inter-monolith arrangement,
and circular concrete cell location. Figure 2.14 shows the plan view and
pile layout for the concrete cell at the start of the guide wall. Figure 2.5 is a
picture of this completed concrete cell. A typical cross section of the
flexible impact deck structural system is presented in Figure 2.15. This
figure also shows a close-up, plan view of a 12 ft 6 in. long precast base
segment. Figure 2.6 is a picture of this precast base segment.

2.4 Lock and Dam 3 - Mathematical Model

Lock and Dam 3 can be considered as a beam element because its length is
much greater than the other two directions. The length is 838.66 ft and the
height and width are 5 ft and 22 ft, respectively. The model can be seen in
Figure 2.19.

ERDC/ITL TR-16-1

Figure 2.13 Lock and Dam 3 guide wall plan view 1 of 2.

iii

SNVId

eiosauujw 'ipRM
lieMapjns wea >8 >pcr|

ERDC/ITL TR-16-1

Figure 2.14 Lock and Dam 3 guide wall plan view 2 of 2.

ERDC/ITL TR-16-1

Figure 2.15 Lock and Dam 3 guide wall detail of pile layout and end cell.

ERDC/ITL TR-16-1

Figure 2.16 Lock and Dam 3 guide wall section view.

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Figure 2.17 Lock and Dam 3 guide wall plan - 5.

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Figure 2.18 Lock and Dam 3 guide wall plan - 6.

ERDC/ITL TR-16-1

The mathematical model can be done using 3-D beam elements. A 3-D
beam element has 6 degrees of freedom per node, producing 12 degrees of
freedom per element. The degrees of freedom per node are 3 translations
and 3 rotations as seen in Figure 2.20. The applied normal force F x (t) is
the impact-force time history developed using the PC-based software
Impact_Force. The applied parallel force F y (t) is a (decimal) fraction of the
normal force calculated using the dynamic coefficient of friction between
the barge and the impact deck surfaces. Note the armor rubbing surfaces
cast into the impact deck face can be seen in detail in Figure 2.10.

Plan View

Figure 2.19 Lock and Dam 3 flexible approach wall.

Lock and Dam 3 Flexible Approach Wall Extension

838'-6 1/8"

Pile

Founded
Concrete
Filled Cell

Barge Train Impact

Pile Group
Supported
Monoliths

Existing

Approach

Wall

1.92" Expansion Joint
ShearConnection
(no moment transfer)

2" Expansion Joint
ShearConnection
(no moment transfer)

1.92" Expansion Joint
(no shear or moment transfer)

Figure 2.20 (a) Typical 3-D segment of the impact-deck beam element, (b) Impact force
applied to the Impact Deck, (c) Typical 3-D beam element.

ERDC/ITL TR-16-1

If the model used to describe the beam is developed in the plane, the beam
element has 3 degrees of freedom per node and 6 degrees of freedom per
element. The degrees of freedom per node are 2 translations and 1
rotation, as seen in Figure 2.21. Based on the notation of Figure 2.20, the
force and moment conditions for node i are Fi, x = Vi, Mi, x = o, Fi, y = Fi, Mi, y
= o, Fi, z = o, and Mi, z = Mi, and for node/are Ff x = Vf, Mf, x = o, Ff, y = Ff, Mf y
= o, Ff, z = o, and Mf z = Mf. Basically, to transform a 3-D beam element to a
2-D (plane element), the moment about the “x” axis, the moment about
the “y” axis, and the force in the “z” directions are equal to zero.

Figure 2.21 Typical 2-D beam element used in lmpact_Deck

The Impact_Deck PC-based computer program is based on beam elements
loaded and deformed in the plane “y-x”. The mathematical model is
presented in Figure 2.22. The connection of the impact deck to the
concrete circular cell is assumed as pinned and the end of the last
monolith is assumed to be free. The inter-monolith connection is
considered as an internal pin where the moment is zero (i.e., no moment
transfer). The normal and shear impact force (time history) is applied
along the beam elements and has a variation in time and position. The
load moves with a constant velocity, so the position of the load varies
linearly with respect to time. The impact deck is supported by equally
spaced nonlinear springs in the global “x” and “y” directions. These
nonlinear springs represent the reaction provided by each row of three
piles through soil-structure interaction with the foundation soil(s).

A description of the beam element used for the first element (i.e., element
that is pinned and connected to the rigid circular cell) is presented in
Figure 2.23. At node 1, which is at the cell to monolith number 8 interface
(left side in Figure 2.19), the element has zero displacements (for these two
DOFs) and also, a zero bending moment. The other force, moment,
displacement, and rotations are not equal to zero.

ERDC/ITL TR-16-1

Figure 2.22 Lock and Dam 3 mathematical model.

Support at circular
rigid structure

L|.'

y, v(y,t), t F x

Inter-monolith pin
connection

j™ j—

b[a[a[a[a[a[a[a[a[a[a[a[a[a[a[a[b

1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ’

Typical first monolith

Inter-monolith pin ^_

connection

_^ Inter-monolith pin

connection

^ r r r r r r r r r r r r r r r

b[ a [a [a [a [a [a [a [a [a [a [a [a [a [a [a [b

1111111111111111'

Typical internal monolith

Inter-monohth pin ^_

connection

•

_^ Beam with

free end

|T r r r r r r r r r r r r r r f

Pi a [a [a [a [a [a [a [a [a [a [a [a [a [a [a [l

’]

1111111111111111

Typical last monolith

Figure 2.23 First beam element used in Lock and Dam 3 mathematical model.

\w, = 0, V,

P 6 h M, = 0

0 2 , M: Y

\ w

? , V 2

Node 1 = Upstream

^ w

Hi = 0 Fi ± _

— W

»i,F,

Node 2= Downstream

^ b

Nodel

Node 2

A description of the beam element used for the last element (i.e., element
adjacent to the existing approach wall) is presented in Figure 2.24. The
force, moment, rotation, and displacement at node (n-i) are similar to node
i of the typical beam element. At node n, located at the monolith number 1
to the existing approach wall interface on the right side in Figure 2.19, the
forces and moment (at node n) are equal to zero (i.e., free beam end).

Figure 2.24 Last beam element used in Lock and Dam 3 mathematical model.

Ui> F l ■

A w,-, Vi

0 i,M t

A WfVf -0

6 f , M f = 0 \ u f F f =0

■+

Node (n-1)

Noden

Node (n-1) = Upstream
Noden = Downstream

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A description of the beam element used for the inter-monolith pin
connection is presented in Figure 2.25. At the inter-monolith pin
connection (node s), the bending moment is equal to zero; but the other
forces, moments, displacements, and rotations are not equal to zero.

Figure 2.25 Description of two beam elements connected at the inter-monolith connection.

A K

0 r , M r

Q s , M s =

_ _ w s , V s
= ° u s ,F s

= 4 —>

Noder

Nodes

Noder = Upstream

Nodes = inter-monolith pin
connection

Node t = Downstream

A w s ,V s

u s ,F s

G S ,M S =0

A w t> v t

G t ,M t 'f\ Uu p t

i —>
-+■

Nodet

Figure 2.26 shows a typical internal monolith and the locations of the
connected inter-monolith beam elements. Observe that the individual pile
group row number (i.e., at the location of each pair of springs) is also
shown in this Figure and labeled as numbers 1 through 16.

Figure 2.26 Beam elements numbering in a typical internal monolith.

The behavior of a single row of three piles under static lateral load was
conducted to determine the force-versus-displacement relationship for the
nonlinear springs used in the Lock and Dam 3 model. A description of the
cross-section of Lock and Dam 3 is presented in Figure 2.27 with the three
batter piles that resist the lateral load. The system has two battered piles
with an inclination of 1 horizontal to 4 vertical and one vertical pile. The
piles extend 24 ft above ground surface and are embedded a vertical
distance of 48 ft into the ground. Each pipe pile is 24 in. in diameter and is
filled with concrete.

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Figure 2.27 Cross-section of Lock and Dam 3.

Two separate push-over analyses were conducted to define the pair of
nonlinear impact deck springs representing the soil-structure interaction
responses in the transverse and longitudinal directions of an individual
batter pile group using the computer program CPGA. The transverse push¬
over result shown in Figure 2.28a is consistent with those reported in the
Ebeling et al. (2012) Figure 4.4 (or Figure B.8). The resulting nonlinear
force-versus-displacement relationship for loading in the transverse
direction is a result of the development of plastic hinging at various
locations within the individual piles, and of the development of (soil-to-
pile interface) tension and/or pile tip (end) bearing soil failures of
individual piles within the batter pile system as the applied lateral load is
increased on the single batter pile group. Details of this push-over analysis
are discussed in section 4 of Ebeling et al. (2012). Figure 2.28a, force-
versus-displacement curve (with a white background, shows that a
horizontal force of 180 kips results in a lateral displacement of 0.2 ft
(2.46 in.); a force of 250 kips results in a lateral displacement of 0.41 ft
(4.9 in.); a force of 272 kips results in a lateral displacement of 0.63 ft
(7.5 in.); and a force of 317.75 kips results in a lateral displacement of 1.1 ft
(13 in.). The data describing this curve is shown as a table in this user
interface (Figure 2.28a). The curve used to define the elastic loading phase
of the load-unload nonlinear process that will be discussed in section 2.5.

ERDC/ITL TR-16-1

Figure 2.28 Transverse and longitudinal push-over results for a
single row of three piles aligned in the transverse direction.

a) Transverse direction

b) Longitudinal direction

The longitudinal push-over results are shown in Figure 2.28b. The
resulting nonlinear force-versus-displacement relationship is a result of
the development of plastic hinging, first occurring at the pile cap and then
at a point along the pile located below the mud line within the individual
piles of the three-pile group as the lateral load is applied to the single
batter pile group in the longitudinal direction (i.e., perpendicular to the
line of piles). The Figure 2.28b force-versus-displacement curve (with a
white background) shows a horizontal force of 123 kips resulting in a
lateral displacement of 0.35 ft (4.18 in.), and a force of 149.7 kips resulting
in a lateral displacement of 0.64 ft (7.6 in.). The curve used to define the

ERDC/ITL TR-16-1

elastic loading phase of the load-unload nonlinear process will be
discussed in section 2.5.

The ultimate push-over capacity of the pile group in the transverse
direction is 317.75 kips as compared to a capacity of 149.7 kips in the
longitudinal direction, a factor of 2.1. For the same level of horizontal
displacement, the push-over curve in the longitudinal direction is of lower
magnitude force than in the transverse direction. This is because for
longitudinal loading each of the three piles responds more like “vertical”
piling under a lateral load. Vertical pile push-over response behavior is
discussed in detail in section 3 of Ebeling et al. (2012). It is observed that
vertical pile behavior under lateral loading occurs without the soil-to-pile
failures and the “pole-vaulting” actions that are unique to a batter pile
group subjected to an applied line of loading in the transverse direction
(i.e., in line with the three-pile group). These push-over results are further
discussed in section 2.9.

2.5 Nonlinear force-deflection relationship for the spring supports

Impact_Deck has the capability to calculate the response of the spring
supports during the time-history analysis even if the springs possess plastic
behavior in their force-displacement relationship of an individual pile
group. The spring is considered as “linear” if the load in the spring is below
the elastic displacement Seias and the elastic force Feias as shown in
Figure 2.29. If the load is reduced, and the force-displacement is below
point 1, the unloading path follows the same path as the previous loading
phase. The loading phase in this figure is depicted as the green arrows and
the unloading phase by the red arrows. However, if the load is greater than
the elastic displacement and is in the loading stage, the load will follow the
path shown using the green arrows until it reaches the maximum force-
displacement value, labeled as point 2 in this figure. If unloading occurs
from this point, it will unload following the user-specified slope that follows
the unload path starting at point 2 and moves in the direction of point 4. If
the pile group nonlinear “spring” is never again subjected to a force as large
as that corresponding to point 2 on this figure, the force-displacement
response will remain along a line from points 2 to 4, until zero force is
reached. However, a plastic permanent deflection equal to the distance from
the origin to point 4, lateral displacements will result for the pile group.
Another scenario is if the load should increase again and go above the point
2 force magnitude, the load-displacement response will follow along the
“original backbone curve” moving from point 2 towards point 3. If the force

ERDC/ITL TR-16-1

reaches a maximum value somewhere between point 2 and 3 and starts to
decrease again, the load-deflection will follow the same unload slope as the
slope between points 2 and 4, but start from the new maximum force-
deflection point. Lastly, if the force-deflection value is greater than that
corresponding to point 3, Impact_Deck assigns a zero value to this spring
because the maximum force value was reached and failure will occur.

Figure 2.29 Force-displacement relation of the spring
support.

2.6 Solving for the motion of the structure

The equations of motion for a flexible approach wall structure comprised of
decks supported on clustered pile groups and their end-release
computations for the Lock and Dam 3 model and other similar structural
systems, is given in Appendix A. Appendix I discusses the Rayleigh damping
feature of the structural model, with section 1.2 giving information specific
to the Lock and Dam 3 model. The numerical methods to be used in the
solution of the equations of motion are either HHT-a or Wilson- 0 , which
are discussed in Appendix F and G, respectively.

2.7 Validation of lmpact_Deck Computer Program

The validation of Impact_Deck computer program using the Lock and Dam
3 model was made against the results obtained from the computer program
SAP2000. The beam for validation has a total length of 838.666 ft long. In
the validation procedure, the beam was modeled with 137 nodes and 136
beam elements. The 7 inter-monolith pin connections (i.e., no bending

ERDC/ITL TR-16-1

moment transfer between adjacent monoliths) were included in the model.
A set of linear springs was located at the node where the pile supports were
placed. 1 The strength of the concrete was assumed as fc = 5,000 psi with a
corresponding modulus of elasticity for the concrete of E = 580,393.25
kips/ft 2 . The beam cross-sectional area and the beam second moment of
area (moment of inertia) were 110.0 ft 2 and 4436.666 ft 4 , respectively. The
mass per linear foot of beam was calculated as m = 0.5127 kip *sec 2 /ft. A
damping factor of 0.02 (i.e., 2% of the critical damping) was used in both
computer program models. The impact-force time history was the Winfield
test # 10 (generated using Impact_Force, Ebeling et al. 2010) and shown in
Figure 2.30. The tangential-force time history was set equal to the
transverse-force time history multiplied by a dynamic coefficient of friction
of 0.5. The impact load was kept stationary at a point 402.896 ft along the
impact deck due to restrictions in loading for SAP2000. The load-deflection
relationship was assumed as the one presented in Figure C-4 of Ebeling et
al. (2012).

Figure 2.31 shows the results obtained for the node where the load was
applied at 402.896 ft along the impact deck. The results are consistent
between the two programs.

1 The SAP2000 analysis is restricted to a linear spring model for each group of clustered piles.

ERDC/ITL TR-16-1

Figure 2.31 Validation of lmpact_Deck against SAP2000.

Response at 402.89 feet
Damping ratio = 2%
-T 0 = 0.408 sec.

— — lmpact_Deck-Linear-v = 0 ft/s — — SAP2000-Linear-v = Oft/s

2.8 Numerical Example of the Elastic-Plastic Response Using
lmpact_Deck

In this section, results from a numerical example are shown that
demonstrate the plastic behavior capability of the nonlinear impact deck
clustered pile springs. Plastic response can develop if the limiting elastic
displacement specified by the user (i.e., Point l in Figure 2.29) is low
enough to force the springs to enter into the zone of plastic response. The
input data for the Impact_Deck computer program for Lock and Dam 3
model included the total length of the beam at 838.666 ft long with 137
nodes and 136 beam elements. The 7 inter-monolith pin connections (i.e.,
no bending moment transfer between adjacent monoliths) were included in
the model. A set of springs was located at the node where the pile supports
were placed. The strength of the concrete was assumed as/c = 5,000 psi
producing a modulus of elasticity for the concrete of E = 580,393.25
kips/ft 2 . The beam cross-sectional area and the beam second moment of
area (moment of inertia) were 110.0 ft 2 and 4436.666 ft 4 , respectively. The
mass per linear foot of beam was calculated as m = 0.5127 kip *sec 2 /ft. A
damping factor of 0.02 (i.e., 2% of the critical damping) was used. The
impact-force time history applied was the Winfield test # 10 as shown in
Figure 2.30. The tangential-force time history was set equal to the
transverse-force time history but multiplied by a dynamic coefficient of
friction of 0.5. In this example, the load was assumed to be in motion along
the impact deck at a velocity of v = 3 ft/sec, starting at the node located at x
= 402.896 ft. The spring stiffnesses assigned in this analysis did not make

ERDC/ITL TR-16-1

use of the Ebeling et al. (2012) Appendix A push-over analysis results for
the 6-ft-diameter vertical piling because the Winfield Test # 10 loads could
not be guaranteed to bring the computed stiffnesses into the zone of plastic
deformation. In an effort to illustrate the effects of plastic deformation, the
force-displacement relationship (backbone curve) was assumed to have the
following slopes (stiffness), ki = 540.0 kip/ft, k2 = 215.0 kip/ft, k3 = 100.46
kip/ft, and the stiffness when it is in the plastic unload path of kunioad =
3-33*ki. The limit for the elastic displacement was assumed as Sdastic =
0.0633 ft. The force-displacement relationship (backbone curve) had a
second break point (second to third slope) at a displacement of 0.30 ft. That
meant that the force value at the inflection point of the slope occured first at
34.182 kips and a second inflection point at 85.073 kips. Figure 2.32 shows
the results obtained for the node where the load was applied at 402.896 ft.
The results in Figure 2.32 show the effect of exceeding the first yield point in
the spring model. The purple values are offset from their original position
by approximately 0.09 ft after the spring force reached its yield point. The
red curve indicates the behavior for a linear elastic spring model where
yielding does not occur. Figure 2.33 shows the plastic behavior of the spring
at 402.896 ft, where the normal impact load had its initial point of contact.
After 3.63 sec, the linear response oscillates around zero displacement and
the plastic response oscillates around 0.098 ft. These behaviors can be
observed in Figure 2.33 where the plastic response was reached (second
slope in the force-displacement diagram), ending with a permanent
displacement of around 0.098 ft.

Figure 2.32 Dynamic response of the transverse spring located at x =

402.896 ft.

Response at 402.89 feet
Damping ratio = 2%
-T 0 = 0.408 sec.

lmpact_Deck-Linear-v = 3 ft/s lmpact_Deck-Nonlinear-v= 3 ft/s

ERDC/ITL TR-16-1

Figure 2.33 Dynamic response of the transverse spring located at x =

402.896 ft.

Nonlinear Force-Displacement
Elastic = 0.0633 ft : kl =540.0 kip/ft :k2 = 215.0 kip/ft :
k3 = 100.46 kip/ft : k unload = 3.33*kj

0 0.2 0.4 0.6 0.8 1 1.2

Lateral Displacement (ft)

2.9 lmpact_Deck GUI results

In section 5, the visualization of data using the Impact_Deck GUI post¬
processing capabilities will be discussed, but these capabilities are being
introduced here to give an idea of the output results from the
Impact_Deck processing code, which used the formulation for impact deck
structures discussed in section 2.8. The results are from the Lock and Dam
3 example problem in section 5. The impact deck geometry and its
material properties were the same as in section 2.8, with the exception of
the load parameters (starting position and velocity along the approach
wall) and the transverse non-linear spring properties of the individual pile
cluster sub-system. The load still used the Winfield Test # 10 impact-force
time history, but the starting position was moved to 501.188 ft from the
beginning of the wall and had a velocity of 1.0 ft/sec (along the wall).

The calculation of pile group stiffness was determined through push-over
analyses performed on a single pile group. The Lock and Dam 3 impact-
deck pile group consisted of a fixed head system of one vertical and two
batter piles, all with a diameter of 24 in. In this case, the push-over
analysis took all the piles into account, including the effects of batter
(Figure 2.34).

ERDC/ITL TR-16-1

Figure 2.34 Reprint of the Figure 4-4 transverse direction of loading
push-over analyses from Ebeling et al. (2012); fixed head results in
green and pinned head results in blue.

The push-over analysis for a transverse load on the fixed head, wet site
analysis (dashed green curve shown in Figures 4-4 and B-8 on pages 81
and 153, respectively, of Ebeling et al. 2012) was defined by the points
listed in Table 2.1. These values were used to define the transverse spring
model for an individual group of 3 piles.

For the longitudinal spring model, the longitudinal forces acting on the
impact deck when the two plastic hinge points occured were 1850 kips and
2970 kips, respectively. These forces were much greater than and likely
from a barge train impact event; thus no yielding of any pile groups in the
longitudinal direction was anticipated. The data contained in Table 2.2
were developed using the push-over analysis procedure outlined in section
3 or Appendix A of Ebeling et al. (2012) for loading applied in the
transverse direction.

ERDC/ITL TR-16-1

Table 2.1 Primary loading curve for the transverse spring model of a single pile group with a leading
vertical pile followed by two batter piles (Lock and Dam 3).

Force

Deflection

Notes on Flexural Plastic Hinging Conditions

(kips)

(inches)

(feet)

From Ebeling et al. (2012)

0.0

180.0

2.462

0.20517

Pile to pile cap moment capacity reached

250.0

4.899

0.40825

Pile 3 yields in axial tension

272.0

7.544

0.62867

Flexural plastic hinges develop in piles below
mudline

317.75

13.044

1.087

Pile 2 buckles

Table 2.2 Primary loading curve for the longitudinal spring model of a single pile group with a
leading vertical pile followed by two batter piles (Lock and Dam 3).

Force

Deflection

Notes on Flexural Plastic Hinging Conditions

(kips)

(inches)

(feet)

0.0

123.0

4.18

0.34833

Pile to pile cap moment capacity reached

149.7

7.62

0.635

Flexural plastic hinges develop in piles below
mudline

According to Figure B-3 of Ebeling et al. (2012), yielding occurred for a
pile when the moment exceeded 8544 kip-inch with no significant axial
loading (i.e., pure bending).

These Table 2.2 values come from the push-over analysis of the 3 pile
system with 2 batter piles that were solved as a system, and therefore, the
transverse and longitudinal push-over analyses must be performed using
CPGA.

The transverse force-displacement relationship (backbone curve) was
therefore assumed to have the following slopes (stiffness), ki = 877.32
kip/ft, k2 = 344.69 kip/ft, k3 = 99.81 kip/ft, and the stiffness when it is in
the plastic unload path of kunioad = i.o*ki. The limit for the elastic
displacement was assumed as deiastic = 0.20517ft. This backbone curve is
shown in Figure 2.28a. This section does not provide an engineering
analysis, but gives an idea of the information provided so that an
engineering analysis might be made.

ERDC/ITL TR-16-1

Nodal outputs provided from the FEO 1 analysis of an impact deck were the
longitudinal displacement, transverse displacement, and rotational
displacement (in radians) for each node at each time step of the simulation.
A Table was also provided that gives the maximum displacements (longi¬
tudinal, transverse, and rotational) for each node and the time that the
maximum displacement occurred.

Figure 2.35 shows the GUI table of maximums for the example problem in
this section. From this GUI table, it is possible to tell the time step and the
node with the maximum displacement for transverse, longitudinal, and
rotational displacements (as shown in Table 2.3).

Figure 2.35 lmpact_Deck GUI Table of maximum nodal displacements
for the L&D3 example impact deck.

* File

1 lmpact_Deck_TestProblem1.feo

® Display extreme values and their times

| © Plot animated [MHIEIIIIHIH]

[l t| Displace

ment/Rotation: |x ▼!

7] Loop Displacement/Rotation:

(*Z3

|0.00000 @| G

Long.

Trans.

Trains.

Rot.

Node

Disp.

Time

Disp.

Time

Disp.

Time

(ft)

(sec)

(ft)

(sec)

(ft)

(sec)

0.138

0.21

0.138

0.21

0.138

0.0001

0.21

0.138

0.0001

0.21

0.136

0.138

0.0001

0.21

0.136

0.0001

0.21

0.132

0.136

0.0002

0.21

0.084

0.136

0.0002

0.21

0.08

0.134

0.0002

0.21

0.112

0.134

0.0002

0.21

0.11

0.132

0.0003

0.21

0.144

0.13

0.0003

0.21

0.144

0.082

0.0003

0.21

0.142

0.08

0.0003

0.21

0.142

0.078

0.0004

0.21

0.142

0.076

0.0004

0.21

0.142

0.076

0.0004

0.21

0.076

0.0004

0.21

0.082

0.108

0.0005

0.21

0.082

0.14

0.0005

0.21

0.082

0.134

0.0005

0.21

0.08

0.13

0.0005

0.21

0.078

0.086

0.0006

0.21

0.078

0.084

0.0006

0.21

0.074

0.082

0.0006

0.21

0.072

0.08

0.0006

0.21

0.068

0.078

0.0007

0.21

0.066

Table 2.3 Maximum nodal displacements for the lmpact_Deck example problem.

Node number

Value

Time (seconds)

Transverse

0.0663 feet

0.26

Longitudinal

137

0.002 feet

0.212

Rotational

0.0008 radians

0.26

1 FEO is the Finite Element Output format for the lmpact_Deck program.

ERDC/ITL TR-16-1

Figures 2.36, 2.37, and 2.38 show the time histories for displacements at
nodes 86,137, and 93, respectively. Because some of these displacements
were very small, and the data were stored with limited precision, some of
these plots developed ‘jaggies’, where the data seems to form stair steps.

Figure 2.36 Transverse nodal displacement time histories for node 86.

ERDC/ITL TR-16-1

Figure 2.38 Rotational nodal displacement time histories for node 93.

The Impact_Deck GUI also allowed the user to visualize the entire beam in
motion using an exaggerated plot of displacements transversely, longitudi¬
nally, and rotationally. Figures 2.39, 2.40, and 2.41 show the displacements
of the wall from these animated plots at the moment where the maximum
displacement occurred, 0.26 sec for longitudinal displacements, 0.21 sec for
transverse displacements, and 0.26 sec for rotational displacements. These
data were also subject to the jaggies because of the precision of the stored
data. The data were scaled to fit the plot.

Element outputs provided from the FEO analysis of an impact deck were
the axial force, shear force, and moment for each element at each time step
of the simulation. A table was also provided that gives the minimum and
maximum forces and moments for each element and the times that the
minimum and maximum force/moment occured.

Figures 2.42 - 2.44 show GUI tables of extreme values for the example
problem in this section. From this GUI table, it is possible to tell the time
step and the element with the extreme axial force, shear force, and
moment (as shown in Table 2.4).

ERDC/ITL TR-16-1

Figure 2.39 Transverse wall displacements at 0.26 sec.

Figure 2.40 Longitudinal wall displacements at 0.21 sec.

ERDC/ITL TR-16-1

Figure 2.42 lmpact_Deck GUI table of element minimum and maximum axial
forces for the L&D3 example impact deck.

ERDC/ITL TR-16-1

Figure 2.43 lmpact_Deck GUI table of element minimum and maximum shear
forces for the L&D3 example impact deck.

® Display extreme values and their times
O Plot force/moment vs. time Element Index; [l -1 Mode: [/teat

O Plot animated \i } || H 0 00000 \rj B Loop Mode: (tad

135

136

- 1.8

-0.45

Shear

Minimum

Force

(kip)

-0.93

-0.91

- 0.86

-0.76

-0.64

-0.5

-0.34

- 0.2

-0.17

-0.25

-0.4

-0.54

- 0.66

-0.75

-0.82

-0.87

-0.91

-0.94

-0.99

-1.04

-1.05

-1

-0.9

-0.76

0.044

0.042

Shear

Minimum

Time

(sec)

0.142

0.14

0.138

0.13

0.118

0.112

0.144

0.142

0.14

0.138

0.136

0.134

0.132

0.13

0.09

0.088

0.086

0.084

1.8

0.45

Shear

Maximum

Force

(kip)

0.93

0.91

0.86

0.76

0.64

0.5

0.34

0.2

0.17

0.25

0.4

0.54

0.66

0.75

0.82

0.87

0.91

0.94

0.99

1.04

1.05

0.9

0.76

0.044

0.042

Shear

Maximum

Time

(sec)

0.142

0.14

0.138

0.13

0.118

0.112

0.144

0.142

0.14

0.138

0.136

0.134

0.132

0.13

0.09

0.088

0.086

0.084

Figure 2.44 lmpact_Deck GUI table of element minimum and maximum
moments for the L&D3 example impact deck.

ERDC/ITL TR-16-1

Table 2.4 Extreme forces/moments for the lmpact_Deck example problem.

Element number

Value

Time (seconds)

Axial

min.

-271.67 kips

0.208

max

528.44 kips

0.200

Shear

min.

-293.15 kips

0.202

max

702.41 kips

0.200

Moment

min.

-5,468.09 kip-feet

0.180

max

5,662.18 kip-feet

0.202

Because the simulated barge train impact occured between nodes shared
by elements 81 and 82 , the extreme forces were found in those elements.
Figures 2.45 - 2.50 show the time histories for the axial force, shear force,
and moments for the elements with the minimum and maximum values,
respectively.

Figures 2.51 - 2.53 show the axial force, shear force, and moment at 0 . 200 ,
0 . 200 , and at 0.180 sec, respectively, for the entire wall. These forces and
moments were collected for animated visualization in the Impact_Deck
GUI.

Figure 2.45 Axial-force time histories for element 82.

ERDC/ITL TR-16-1

Figure 2.46 Axial-force time histories for element 81.

FEO Element Output

• Impact JDeck_TestProbleni1 .feo

] Mode: | Axial ▼]

^j| 0 Loop Mode: [Axial

Axial Force

ERDC/ITL TR-16-1

Figure 2.49 Moment time histories for element 81.

FEO Element Output

lmpact_Deck_Test Problem 1 _pushover.feo

® Plot fbrce/moment vs. time Element Index [81 Mode: [ Moment -• |

Plot animated ^ || ^ C.3C333

Moment (kip-ft)

moment (kip

ERDC/ITL TR-16-1

Figure 2.50 Moment time histories for element 82.

File

: lmpact_Deck_Test Problem 1 _pusho ver.f eo

1 © Display extreme values and their times

1 (®) Plot force/moment vs. time Element Index: [82

▼ | Mode: |Moment

Plot animated ^ || ^ M ;

[4JI13 Loop Mode: |Axial

Elem = 82

Moment (kip-ft)

Figure 2.51 Wall axial forces at 0.2 sec.

ERDC/ITL TR-16-1

Figure 2.53 Wall moments at 0.18 sec.

Of primary importance for doing a pile group founded flexible wall analysis,
is being able to see exactly how much force each pile group will be able to
resist during an impact event. This is measured by finding the resisting
force from the spring model used for the pile group for the pile group’s

ERDC/ITL TR-16-1

displacement. Figure 2.54 shows the table of maximum forces/moments
resisted at the nodes representing a pile group in the impact deck example
model. Looking through this table gives the information that the maximum
individual pile group response force is in the transverse direction at node 85
and at time 0.26 sec. This peak response force is 54.465 kips. Figure 2.55 is
the same window scrolled down to reveal the maximum displacements of
these pile group nodes, and when these displacements occurred.

A second table in the program reveals the response forces of each pile
group node at a specified time. Figure 2.56 shows this table for time
0.26 sec. At the bottom of this table, the individual responses are summed
to reveal the total force in each direction. Information about the impulse
calculations over all time is also given at the end of this table. For time
0.26 sec, the total force from the 96 pile groups acting in the transverse
direction is 722.776 kips. For the entire run, the transverse impulse was
454.31 kip-sec.

Figure 2.57 shows the response of the pile group at node 85 at time
0.260 sec as a force-versus-displacement plot, and as force and
displacement time histories. This is the time when the displacement for
the pile group at node 85 is at its maximum location in the transverse
direction. In this example, the longitudinal and rotational displacement
created small resisting forces (Figures 2.58 and 2.59).

Figure 2.54 Table of pile group response maximum forces and moments and their

time.

Pile Group Response

File

lmpact_Deck_TestProblem1.feo

0 Display extreme values and their times

O Display Spring Forces at Time: 0.0000

- Plot Animated IMIIHII ► IIIIIIMIIMI JoSOQOO

[3 Loop Pile Group

Node Index [ 1

▼ | Degree of Freedom: |x ▼ !

Long.

Trans.

Node

Force

Time

Force

Time

Moment

Time

(kips)

(sec)

(kips)

(sec)

(kip-ft)

(sec)

251.4489

0.21

0.9333

0.142

0.0012

0.0047

0.21

0.001

0.18

0.0138

0.21

0.0028

0.18

0.0229

0.21

0.0045

0.18

0.032

0.21

0.006

0.18

0.0411

0.21

0.0073

0.178

0.0502

0.21

0.0083

0.178

0.0593

0.21

0.009

0.178

0.0684

0.21

0.0093

0.178

0.0775

0.21

0.0093

0.178

0.0866

0.21

0.0089

0.176

0.0957

0.21

0.0082

0.176

0.1048

0.21

0.0078

0.086

0.1139

0.21

0.0082

0.082

0.123

0.21

0.0088

0.08

0.1321

0.21

0.0096

0.078

0.1412

0.21

0.0105

0.076

0.1505

0.21

0.0092

0.076

0.1596

0.21

0.0059

0.072

0.1687

0.21

0.0059

0.14

0.1778

0.21

0.007

0.134

0.1869

0.21

0.0087

0.13

0.196

0.21

0.0105

0.126

0.2051

0.21

0.0121

0.124

0.2143

0.21

0.0134

0.124

0.2234

0.21

0.0144

0.122

0.2325

0.21

0.0148

0.12

0.2416

0.21

0.0149

0.118

ERDC/ITL TR-16-1

Figure 2.55 Table of pile group response maximum displacements and their time.

lmpact_Deck_TestProbleml .feo

. o Display extreme values and their times

1 O Display Spring Forces at Time: 0.0000

O Plot Animated M M ► II N 0.00000

0 Loop Pile Group Node Index [l

▼ | Degree of Freedom: [x

134

0.7024

0.212

0.0669

0.214

135

0.7024

0.212

0.0247

0.214

136

0.7024

0.212

0.0029

0.212

Long.

Trans.

Node

Disp.

Time

Disp.

Time

Rot

Time

(in)

(sec)

(in)

(sec)

(rad)

(sec)

IE-05

0.08

4E-05

0.162

6E-05

0.152

IE-05

0.178

9E-05

0.172

IE-05

0.094

0.00012

0.196

IE-05

0.09

0.00014

0.176

IE-05

0.086

0.00017

0.192

IE-05

0.084

0.00019

0.178

IE-05

0.082

0.00022

0.19

IE-05

0.08

0.00025

0.208

IE-05

0.076

0.00027

0.19

IE-05

0.074

0.0003

0.202

IE-05

0.07

0.00032

0.188

IE-05

0.068

0.00035

0.198

IE-05

0.064

0.00037

0.188

IE-05

0.062

0.0004

0.196

IE-05

0.06

0.00041

0.19

IE-05

0.058

0.00045

0.194

IE-05

0.062

0.00048

0.2

IE-05

0.13

0.0005

0.192

IE-05

0.122

0.00053

0.198

IE-05

0.116

0.00056

0.21

IE-05

0.112

0.00058

0.196

IE-05

0.11

Figure 2.56 Table of pile group responses for each pile group individually and
summed (not shown) at time 0.26 sec.

Pile Group Response

File

| lmpact_Deck_TestProbleml.feo

|| © Display extreme values and their times

II ® Display Spring Forces at Time: jl2600

|l © Plot Animated M M ► II N HI 100000

[£-J| 0 Loop Pile Group Node Index [l

▼ | Degree of Freedom: [x

[Time:

0.26

Long.

Trans.

Node

Force

Moment

(kips)

(kip-

ft)

- 0.302

0.0041

0.0003

0.0121

0.0008

0.0201

0.0013

0.0281

0.0018

0.0361

0.0021

0.0442

0.0024

0.0522

0.0025

0.0602

0.0024

0.0682

0.0022

0.0762

0.0019

0.0842

0.0014

0.0922

0.0008

0.1002

0.0001

0.1082

- 0.0007

0.1163

- 0.0016

0.1243

- 0.0025

0.1325

- 0.0029

0.1405

- 0.0027

0.1485

- 0.0025

0.1565

- 0.0022

0.1645

- 0.0018

0.1725

- 0.0013

0.1805

- 0.0006

0.1886

0.0002

0.1966

0.0011

ERDC/ITL TR-16-1

Figure 2.57 Transverse pile group responses for the pile group at node 85 and at

time 0.24 sec.

Figure 2.58 Longitudinal pile group responses for the pile group at node 85 and

at time 0.24 sec.

File

lmpact_Deck_TestProbleml feo

® Plot Animated [MlfflfFlffllfflBl] [026000 fjj| m Loop Pile Group Node Index [85 Degree of Freedom: [y

Time = 0.26000 Node: 85 Y-Axis

ERDC/ITL TR-16-1

Figure 2.59 Rotational pile group response for the pile group at node 85 and at

time 0.24 sec.

The Case for Dynamic Analysis:

This example problem demonstrated the need to perform a dynamic
analysis of these pile founded walls. This example used an impact time
history that was the result of the Winfield Test # 10. This impact time
history was applied transverse to the approach wall at a position starting
at 501.187 ft along the wall (at the same location as node 82) and moved at
1 ft/sec along the approach wall. The peak force for the impact time history
was 516.4 kips at time 0.174 sec.

Node 85 (as discussed above) had a maximum transverse force of 56.4652
kips, which occurred at 0.26 sec. This was the last pile group node along a
monolith, so its transverse displacement was affected by the moment
induced (to the monolith it is connected) by the impact load. The total pile
group response for the whole impact deck structure (96 pile group nodes)
at time 0.26 sec was 722.775 kips, which was greater than the peak input
force. It was also the peak overall response.

At time 0.174, the impact load reached its first peak with a force of
516.2 kips. Data was also collected at this time. Table 2.5 shows the values
generated by this data collection. Figure 2.60 takes this information a step
further by displaying the time histories for the three forces.

ERDC/ITL TR-16-1

Table 2.5 Transverse forces with respect to time.

Time (s)

Node 85 Force (kips)

Impact Load Force
(kips)

Total Pile Group
Response (kips)

0.174

39.131

516.200

324.254

0.200

47.277

516.400

611.247

0.260

56.465

446.358

722.776

Figure 2.60 Time-history plot of transverse input forces, total force response for
all the pile groups, and an individual pile group response forces.

From Table 2.5 and Figure 2.60, it can be seen that the overall pile group
response is dynamic because it does not track with the input force.

Instead, the pile groups respond to the input force overtime due to inertial
effects. Because of this, the overall pile group response can have higher
peak forces. For example, at a time of peak response of 0.2 sec, a
maximum impact force of 516.4 kips is applied to the approach wall. The
total transverse-force response of 96 nodes represents all of the pile group
and totals 611.25 kips. This 18 % larger force response is due to the
contribution of the first two terms of the equation of motion (also seen in
Appendix A) for the dynamic structural response:

M{ii(t)} + [C]{u(t)} + [Jf]{u (0} = {-*=■(»)}

(2.1)

ERDC/ITL TR-16-1

At time 0.26 sec, when a peak response force is recorded at node 85 for an
individual pile group, the contribution of the first two terms of the
equation of motion for the dynamic structural response is even larger
resulting in an overall response force of 722.776 kips. This is nearly 40%
greater than the peak input force of 516.4 kips. And an overall peak
response force of 722.776 kips is 55% larger than the 446.358 kips input
force imposed at 0.26 sec. These observations demonstrate the importance
of applying the equation of motion for calculating pile group structural
response forces (and displacements). These differences explain why a
dynamic analysis is required versus a static analysis, which the user
provided impact load is applied as a single peak value (e.g., determined to
be the input peak force from the time history).

Load Sharing in Dynamic Analysis:

Another point to note is the individual pile group response has much lower
peak forces than the input load. This is because the load is being shared
with other pile groups along the 96 groups that support the eight impact
deck monoliths, which sum up to the overall pile group response. The peak
response of node 85, at 56.465 kips, is only 11% of the peak input force of
516.4 kips and 8% (1/12) of the overall peak response force of 722.776 kips.
Because there are 16 pile group nodes per monolith, it may be surmised
that the peak individual response force would be 1/ 16 th instead of 1/ 12 th of
the overall response. Note the fact that this peak force is obtained only at
node 85 and includes transverse displacement due to rotational moment
and inertia, which are difficult to predict without dynamic analysis.

Validation Using Impulse Calculations:

The higher peak values of the overall pile group response seem out of place
until an impulse calculation (taking the area beneath the time history
curves for input load and overall pile group response in Figure 2.60) is
performed. Despite inertial effects, the impulse of the input load must be
equivalent to the overall pile group response, if the piles do not fail. When
an impulse calculation is performed for the overall pile group response,
the result is 454 kip-sec. The impulse for the input force is 463 kip-sec.

The difference is minimal (<2%) and easily explained by the fact that these
are only transverse forces and do not include moment arm effects that are
induced by the impact load on the impacted monolith.

ERDC/ITL TR-16-1

2.10 Final Remarks

In this section, the Lock and Dam 3 physical model was presented and the
mathematical model to calculate the dynamic response was also
developed. Impact_Deck, a computer program, was used to calculate the
dynamic response of an elastic beam supported over linear elastic or
plastic spring supports. The mathematical formulation featured a method
to calculate the end release at the inter-monolith connection. The impact
normal and parallel concentrated external load was located at a specified
location or assumed to have motion at a specified constant velocity. The
damping effect was considered by means of the Rayleigh damping model,
which depended on the natural frequencies of the system. These natural
frequencies were calculated in an approximate way by using the linear
stiffness of the springs and the mass per unit length of the beam. The
results of Impact_Deck proved to be valid when compared to the results
obtained with SAP2000. An example was presented to show the plastic
behavior of the springs and how these results compared to the linear
elastic response.

ERDC/ITL TR-16-1

3 An Approach Wall with Impact Beams on
Nontraditional Pile Supported Bents -
McAlpine Example

3.1 Introduction

This chapter summarizes an engineering methodology and corresponding
software feature contained within the Impact_Deck software for performing
a dynamic structural response analysis of a flexible impact beam supported
over groups of clustered vertical piles configured in a triangle pattern and
subjected to a barge impact event. This example is based on an alternative
approach wall design proposed for the McAlpine Locks and Dam on the
Ohio River in Kentucky.

3.2 Alternative Flexible Approach Wall - Physical Model

McAlpine alternative flexible approach wall is located in Louisville,
Kentucky, at the Falls of the Ohio. McAlpine Locks and Dam are located at
mile point 606.8 and control a 72.9-mile-long (117.3 km) navigation pool.
The McAlpine locks underwent a 10-year expansion project that was
completed in early 2009. The flexible approach wall system discussed in
this section was an alternative flexible approach wall design.

An artist rendering of the McAlpine alternative flexible approach wall (plan
and cross-sectional views) is shown in Figure 3.1. The structure consists of
continuous, flexible concrete beams with a span of approximately 96 ft per
segment. Each end of the beam rests on a pile cap that is supported by a

ERDC/ITL TR-16-1

group of three piles. The supporting piles provide a flexible resistance to the
barge train impact forces that are applied to the beam. The continuity of the
impact beam with the pile cap impact feature is achieved by means of shear
key at each pile group support. That means that the ends of each beam
transfers the longitudinal and transverse forces with no moment transfer to
each pile group support (i.e., a pinned support). The axial and transverse
forces at the end of the beams are transferred to the pile cap by means of a
shear key. The shear key portion of the pile cap structural feature has a
length of 11.5 ft. The length of the shear key is the distance between two
consecutive concrete beams in this figure. The massive pile cap rests over a
clustered group of three vertical piles. Each pile is 5 ft 8 in. in diameter.
They are arranged in a triangular pattern so as to absorb the torsion
generated at the pile group due to the eccentricity between the center of the
pile group and the location of the end of each of the flexible impact beams
that are supported by a clustered pile group.

Figure 3.1 McAlpine alternative flexible approach wall.

3.3 McAlpine Alternative Flexible Approach Wall - Mathematical
Model

The McAlpine alternative flexible approach wall can be modeled using beam
elements because its length is much greater than its other two dimensions.
The length of each flexible impact beam segment is approximately 96 ft and
the width and height are 6 ft 7 in. and 9 ft, respectively. The mathematical
model can be seen in Figure 3.2. In the analysis, the model consists of two
consecutive beams with longitudinal and transverse elastic-plastic spring

ERDC/ITL TR-16-1

supports at the start and end of the system. These two set of springs model
the pile group at the start and end of the two consecutive beams. The
transverse non-linear spring represents the effect of the clustered group of
three vertical piles. The longitudinal end spring represents the effect of the
response of the end support pile group and that of the other groups of piles
beyond this location. At the center pile group, three rigid links are used to
model the high stiffness and mass of the pile cap. The nodes that connect
the impact beams to the pile cap do not transfer moment between these
impact beams and the center pile cap. The center pile group is modeled with
three springs (two translational and one rotational). The translational
spring stiffness is calculated by means of a push-over analysis (section 3 or
Appendix A in Ebeling et al. 2012) and the rotational spring stiffness is
calculated and seen in Appendix E.

Figure 3.2 McAlpine flexible approach wall mathematical model.

The mathematical model is formulated using 3-D beam elements. A 3-D
beam element has 6 degrees of freedom per node, producing 12 degrees of
freedom per element. The degrees of freedom per node are 3 translations
and 3 rotations as shown in Figure 3.3. The force F x (t) applied normal to
the flexible beams is the impact-force time history developed using the PC-
based software Impact_Force (Ebeling et al. 2010). The force F y (t) is
applied parallel to the wall and is a fraction of the normal force calculated
using the dynamic coefficient of friction between the barge and the flexible
beam contact surface in the Impact_Deck software.

The model used to describe the beam is developed in the plane so the
beam element has 3 degrees of freedom per node and 6 degrees of freedom
per element. The degrees of freedom per node are 2 translations and 1

ERDC/ITL TR-16-1

rotation (Figure 3.4). Based on the notation of Figure 3.3, the force and
moment conditions for node i are Fi, x = Vi, Mi, x = o, Fi, y = Fi, Mi, y = o, Fi, z =
o, and Mi, z = Mi, and for node/ are Ff x = Vf, Mf, x = o, Ff, y = Ff, Mf, y = o, Ff, z =
o, and Mf z = Mf. To transform a 3-D beam element to a 2-D (plane
element), the moment about the “x” axis, the moment about the “y” axis,
and the force in the “z” directions are equal to zero.

Figure 3.3 (a) Typical 3-D segment of the Impact Deck beam element, (b) Impact force applied
to the Impact Deck, (c) Typical 3-D beam element.

Figure 3.4 Typical 2-D beam element used in lmpact_Deck.

e u M t

U j,

Node i

f w fi V f

Nodef u _i F f

A push-over type of analysis was conducted to characterize the force-
versus-displacement behavior of a cluster of three vertical piles under
static lateral loading and thus define the stiffness coefficient of the two
translational springs and one rotational spring. Appendix E summarizes
the three-spring stiffness for the McAlpine alternative flexible wall.

3.4 Nonlinear force-deflection relationship for the springs supports

Section 2.5 discussed how the nonlinear spring response force-
displacement backbone curve was used to model plastic deformation in
the pile substructure for dynamically loaded structures. Push-over

ERDC/ITL TR-16-1

calculations were similarly performed for the pile layout for the McAlpine
alternative flexible wall supports.

The McAlpine alternative flexible wall bent was different from the other
bents, in that its piles were not in a straight line. Its drilled-in-place (DIP)
piles were placed in a triangle (per the following figure), and it was
assumed that a push-over analysis would yield different results than for a
traditional pile bent.

Figure 3.5 Plan view of the flexible wall pile layout.

After a push-over analysis in CPGA was performed, it was found that this
was not the case, other than the fact that due to the size of the beams and
the method of anchoring the beams to the pile cap superstructure the
bents actually developed bending failure at the cap and mudline
simultaneously.

The transverse and longitudinal push-over, force-versus-displacement
curves were effectively the same as taking an individual pile curve and
multiplying the force by three. Tables 3.1 and 3.2 summarize the primary
loading curves used to define the transverse and longitudinal spring
models, respectively.

Appendix E provides a method for calculating the rotational force-versus-
displacement curve from the force-displacement curve for a single pile.
The single pile curve is provided in Table 3.3.

ERDC/ITL TR-16-1

Table 3.1 Primary loading curve for the transverse spring model for a McAlpine alternative flexible wall

bent (3 piles).

Force

Deflection

Notes

(kips)

(inches)

(feet)

Adapted from Appendix A of Ebeling et al.
(2012)

0.0

2141.5

1.8

0.15

Pile to pile cap moment capacity reached
at the same time hinge develops at
equivalent depth of fixity

2141.5

4.0

0.33

Plastic hinge rotation

Table 3.2 Primary loading curve for the longitudinal spring model for a McAlpine alternative flexible

wall bent (3 piles).

Force

Deflection

Notes

(kips)

(inches)

(feet)

Adapted from Appendix A of Ebeling et
al. (2012)

0.0

2141.5

1.8

0.15

Pile to pile cap moment capacity
reached at the same time hinge
develops at equivalent depth of fixity

2141.5

4.0

0.33

Plastic hinge rotation

Table 3.3 Primary loading curve for a spring model for a single 6-ft diameter DIP pile.

Force

Deflection

Notes

(kips)

(inches)

(feet)

Adapted from Appendix A of Ebeling et
al. (2012)

0.0

713.8

1.8

0.15

Pile-to-pile cap moment capacity
reached at the same time hinge
develops at equivalent depth of fixity

713.8

4.0

0.33

Plastic hinge rotation

3.5 Solving for the motion of the structure

The equations of motion for a flexible approach wall structure comprised
of decks supported on clustered pile groups and their end-release
computations for the McAlpine alternative flexible approach wall model
and other similar structural systems are given in Appendix B. Appendix I
discusses the Rayleigh damping feature of the structural model, with
section I.3 giving information specific to the McAlpine alternative flexible
approach wall model. The numerical methods to be used in the solution of
the equations of motion are either HHT-a or Wilson- 0 , which are
discussed in Appendix F and G, respectively.

ERDC/ITL TR-16-1

3.6 Validation of lmpact_Deck Computer Program

The validation of Impact_Deck computer program for the McAlpine
alternative flexible approach wall model was made against the results
obtained from the computer program SAP2000. The beam had a total
length of 180.5 ft long. In the validation procedure, the beam was modeled
with 24 nodes and 24 beam elements. The three rigid elements that model
the pile cap support were included in the model. A set of linear elastic
springs was located at node 1 and 23 where the start and end pile supports
were placed. 1 The strength of the concrete was assumed as fc = 5,000 psi
with a corresponding modulus of elasticity for the concrete of E =
580,393.25 ksf. The beam cross-sectional area and the beam second
moment of area (moment of inertia) were 54.668 ft 2 and 517.2 ft 4 ,
respectively. The mass per linear foot of beam was calculated as in =

0.25486 kip *sec 2 /ft. A damping factor of 0.02 (i.e., 2% of the critical
damping) was used in both computer program models. Figure 3.6, impact-
force time history was the Winfield test # 10 (generated using Impact_
Force, Ebeling et al. 2010) and applied at node 11 (i.e., x = 84.5 ft). The
tangential-force time history was set equal to the transverse time history
multiplied by a dynamic coefficient of friction of 0.5. The impact load was
kept stationary at node 11 due to restrictions in loading for SAP2000. A pile
group rotational spring stiffness value of 12,426,909 kip*ft/rad was com¬
puted as outlined in the Appendix E calculation steps. This calculation used
a stiffness value of 132,189.5 kip/ft as the initial segment of the push-over
curve which was applied to each of the two translational springs for each
pile of the three-pile groups. The stiffness of the three (two translational and
one rotational) representing the central pile group was assigned to node 12’,
located 7.708 ft in the transverse direction behind node 12. This position
was located at the center of rotation of the pile group. Figures 3.7 to 3.10
show the dynamic response time histories of nodes 1,12’, and 23. With the
exception of a few minor differences, both computer programs resulted in
the computation of essentially the same system response values. At nodes 1
and 23, some differences in the magnitude of the computed transverse dis¬
placement values were apparent. However, the magnitudes of these values
were very small, which was associated with numerical approximations.
Figure 3.10 shows the rotation time histories computed at nodes 12 and 12’.
These two nodes defined the rigid beam element of the pile cap, which was
perpendicular to the impact beam model. The rotations were the same for
both nodes and indicated a rigid element behavior.

1 The SAP2000 analysis is restricted to a linear spring model for each group of clustered piles.

ERDC/ITL TR-16-1

Figure 3.6 Force time history of Winfield Test # 10.

- Normal Force - Parallel Force

Figure 3.7 Validation of lmpact_Deck against SAP2000 - Transverse
displacement at node 1.

ImpactDeck - Linear - Linear - Elastic - v = 0 ft/s
Flexible Wall - X - Direction
(No moving load applied at node 11)

(kr)l = 12,426,909 : (kr)2 = 12,426,909

Time (sec)

-ImpactDeck-Node 1 • • • • SAP2000-Node 1

ERDC/ITL TR-16-1

Figure 3.8 Validation of lmpact_Deck against SAP2000 - Transverse
displacement at node 23.

ImpactDeck - Linear - Elastic - v = 0 ft/s
Flexible Wall - X - Direction
(Mo moving load applied at node 11)

(kr)l = 12,426,909 : (kr)2 = 12,426,909

Time (sec)

-Impact_Deck-Node 23 -SAP2000-Node 23

Figure 3.9 Validation of lmpact_Deck against SAP2000 - Transverse
displacement at node 12’.

Impact_Deck

Flexible Wall - X - Direction
(kr)l = 12,426,909 : (kr)2 = 12,426,909

Time (sec)

- Impact_Deck-Node 12' - v=0 ft/s - Linear - SAP2000 -Node 12 - v=0 ft/s - Linear

— SAP2000-\ode 12' - v=0 ft/s - Linear

ERDC/ITL TR-16-1

Figure 3.10 Validation of lmpact_Deck against SAP2000 - Rotation at

node 12 and 12’.

Flexible Wall

Node 12 - 12' - Rotation - Z

(Load applied at node 11 moving at the specified velocity "v")
(kr)l = 12,426,909 : (kr)2 = 12,426,909

Time (sec)

- Impact_Beam-Node 12 - v=0 ft/s - Linear • • • • Impact_Deck-Node 12' - v=0 ft/s - Linear

- SAP2000-Node 12 - v=0 ft/s - Linear - SAP2000-Node 12' - v=0 ft/s - Linear

3.7 Numerical Example of the Elastic-Plastic Response Using
lmpact_Deck

This section presents the results of a numerical example that demonstrates
the plastic behavior capability of the nonlinear impact-deck clustered pile
springs. Plastic response can develop if the limiting elastic displacement
specified by the user (i.e., Point l in Figure 3.5) is low enough to force the
springs to enter into the zone of plastic response. The input data for the
Impact_Deck computer program for McAlpine alternative flexible approach
wall were as follows: The beam has a total length of 180.5 ft long with 24
nodes and 24 beam elements. The three rigid elements that model the pile
cap support was included in the model. A set of linear elastic springs were
located at node 1 and 23 where the start and end pile supports were placed.
The strength of the concrete was assumed as fc = 5,000 psi producing a
modulus of elasticity for the concrete of E = 580,393.25 ksf The beam
cross-sectional area and the beam second moment of area (moment of
inertia) were 54.668 ft 2 and 517.2 ft 4 , respectively. The mass per linear foot
of beam was calculated as in = 0.25486 kip *sec 2 / ft. A damping factor of
0.02 (i.e., 2% of the critical damping) was used. The force time history was
the Winfield test # 10 and shown in Figure 3.6. The tangential-force time
history was set equal to the transverse force time history but multiplied by a
dynamic coefficient of friction of 0.5. In this example, the load was assumed
to be in motion along the impact beam at a velocity of v = 3 ft/sec starting at
the node located at x = 84.5 ft in one set of calculations and applied at x =
84.5 ft along the beam for the entire duration of the impact event in the

ERDC/ITL TR-16-1

second set of calculations (i.e., with at v = oft/sec). In order to ensure
plastic deformations, the spring models obtained by the push-over analysis
were not used. Instead, the initial slopes for the stiffness of the two
translational springs were assigned values equal to fa = 132,189.5 kip/ft and
k 2 = 66,094.75 kip/ft with a stiffness for unload after the elastic
displacement equal to kunioad = ki. The elastic displacement that defines the
point of demarcation for elastic and plastic zone behavior was deiastw = 0.003
ft. The rotational spring stiffness was set equal to far = 12,426,909.0
kip*ft/rad and far = 6,213,454.5 kip*ft/rad with a stiffness for unload after
the elastic displacement was equal to ( fanioadf = far. The elastic rotation that
defines the point of demarcation for elastic and plastic zone behavior was
Qelastic = 0.25 rad. The stiffness of the three (two translational and one
rotational) representing the central pile group was assigned to node 12’,
located 7.708 ft in the transverse direction behind node 12. Figure 3.11
shows the dynamic response time history obtained for node 12 and 12’ (i.e.,
center of rigidity of the three pile group). The green nodal displacement
trace in this figure shows a permanent lateral displacement of approxi¬
mately 0.0011 ft after the transverse pile group spring develops plastic
behavior. Figure 3.12 shows the rotational behavior of node 12 and 12’. If the
impact load is moving, the response shows a change in sign for the rotation
indicating that the load moves from one rigid element to the next rigid
element, that is, the load moves from the right element from the centerline
of the pile cap to the left element. After 3.63 sec, the linear response
oscillates around zero displacement and the plastic response oscillates
around 0.0011 ft. The explanation for the noted behaviors is explained by
the fact that Figure 3.13 shows that plastic response is reached (i.e.,
response along the second slope of the force-displacement diagram) ending
with a permanent displacement of around 0.0011 ft.

3.8 lmpact_Deck GUI results

The Impact_Deck GUI was also used to run the McAlpine flexible wall
problem in this section. This section does not provide an engineering
analysis, but gives an idea of what information is provided so that an
engineering analysis might be made.

The inputs are the same as those used for validating the model, with the
minor exception that the longitudinal velocity of the barge train was 1 ft/sec,
and that the push-over analysis spring models for the pile groups were used.
These changes did not allow the pile groups to go into plastic deformation
during loading.

ERDC/ITL TR-16-1

Figure 3.11 Dynamic transverse response of node 12 and 12’.

ImpactDeck

Flexible Wall - X - Direction

(Load applied at node 11 moving at the specified velocity "v")
(kr)l = 12.426.909 : (kr)2 = 12,426,909
(kr)l = 12,426,909 : (kr)2 = 6213.454.52

Time (sec)

- Impact_Deck-Node 12' - v=0 ft/s - Elastic - Impact_Deck-Node 12' -v=3 ft/s - Elastic

- Impact DeckNode 12' - v=3 ft/s - Plastic ^“Load Scaled

Figure 3.12 Dynamic response of the rotational spring at node 12 and 12’.

Flexible Wall

Node 12 - 12' - Rotation - Z

(Load applied at node 11 moving at the specified velocity "v")
(kr)l = 12,426,909 : (kr)2 = 12,426,909
(kr)l = 12,426,909 : (kr)2 = 6213.454.52

Time (sec)

- Impact_Beam-Node 12 - v=0 ft/s - Linear • • • • Impact Deck-Node 12' - v=0 ft/s - Linear

-Impact Deck-Node 12' - v=3 ft s -Linear _ _Impact Deck-Node-12' - v=3 ft s - Nonlinear

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Figure 3.13 Dynamic response of the transverse spring located at x =

84.5 ft.

No n lin ea r Fo rce-D isp la cemen t

^elastic = 0 003 ft : kl = 132,189.5 kip/ft: k2 = 66,094.75 kip/ft : k nnhMld = k 1

0 0.002 0.004 0.006 0.008 0.01

Lateral Displacement (ft)

Nodal outputs provided from the FEO analysis of a flexible wall are the
longitudinal displacement, transverse displacement, and rotational
displacement (in radians) for each node at each time step of the simulation.
A table is also provided that gives the maximum displacements
(longitudinal, transverse, and rotational) for each node and the time that
the maximum displacement occurs.

Figure 3.14 shows the GUI table of maximums for the example problem in
this section. From this GUI table, it is possible to tell the time step and the
node with the maximum displacement for transverse, longitudinal, and
rotational displacements (as shown in Table 3.4).

Figures 3.15, 3.16, and 3.17 show the time histories for displacements at
nodes 21 and 22 (longitudinal and rotational). Because some of these
displacements are very small, and the data were stored with limited
precision, some of these plots develop jaggies, where the data seems to
form stair steps.

The Impact_Deck GUI also allows the user to visualize the entire beam in
motion using an exaggerated plot of displacements transversely, longitudi¬
nally, and rotationally. Figures 3.18,3.19, and 3.20 show the displacements
of the wall from these animated plots at the moment where the maximum
displacement occurred: 0.252 sec for transverse displacements, 0.192 sec
for longitudinal displacements, and 0.22 sec for rotational displacements.
These data were also subject to the jaggies because of the precision of the
stored data. The data were scaled to fit the plot.

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Figure 3.14 lmpact_Deck GUI table of maximum nodal displacements for the

McAlpine flexible wall.

Table 3.4 Maximum nodal displacements for the McAlpine flexible wall example problem.

Node number

Value

Time (seconds)

transverse

0.0659 feet

0.252

longitudinal

0.0016 feet

0.192

rotational

0.0042 radians

0.220

Figure 3.15 Transverse nodal displacement time histories for node 21.

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Figure 3.16 Longitudinal nodal displacement time histories for node 22.

FEO Nodal Output

File

FlexibleWallTestl .feo

® Plot displacement vs. time Node Index 122

▼ Displacement/Rotation: |y

|4j| 0 Loop Displacement/Rotation: |x ▼ !

Y »

Node =22 Y Displacement

-0.0007

disp. (ftj

Figure 3.17 Rotational nodal displacement time histories for node 22.

FEO Nodal Output

1 Rle

" FlexibleWallTestl.feo

1 © Display extreme values and their times

| (®) Plot displacement vs. time Node Index 22

▼ Displacement/Rotation: |z ▼

I © Plot animated Hj[MlfyifTT(HlWll I 0 - 00000

|4j| 0 Loop UisplacementfRotation: |x »|

Y •

Ll:

2.0 2.5

time (s)

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Element outputs provided from the FEO analysis of an impact deck are the
axial force, shear force, and moment for each element at each time step of
the simulation. A table is also provided that gives the minimum and
maximum forces and moments for each element and the times that the
minimum and maximum force/moment occur.

Figure 3.21 shows the GUI tables of extreme values for the example
problem in this section. The beginning of the axial force extremes table is
shown. The shear force extremes table and moment extremes table are
available by scrolling in the interface. From this GUI table, it is possible to
tell the time step and the element with the extreme axial force, shear force,
and moment (as shown in Table 3.5).

Because the simulated barge train impact occurs between nodes shared by
elements 20 and 21, the extreme forces are found primarily in those
elements. Figures 2.44 - 2.49 show the time histories for the axial force,
shear force, and moments for the elements with the minimum and
maximum values, respectively.

Figures 3.28 - 3.30 show the axial force at 0.2 sec, shear force at 0.22 sec,
and moment at 0.220 sec for the entire wall. These are created from the
Impact_Deck GUI feature to view the animated forces.

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Of primary importance for doing the pile group founded flexible wall
analysis, is being able to see exactly how much force each pile group will be
able to resist during an impact event. This is measured by finding the
resisting force from the spring model used for the pile group for the pile
group’s displacement. Figure 3.31 shows the table of forces/moments
resisted at the nodes representing a pile group in the Impact-Deck
example model.

Figure 3.21 lmpact_Deck GUI table of element minimum and maximum axial
forces for the McAlpine flexible wall example.

Table 3.5 Extreme forces/moments for the lmpact_Deck example problem.

Element number

Value

Time (seconds)

Axial

min

-332.80 kips

0.200

max

332.80 kips

0.200

Shear

min

-541.18 kips

0.220

max

541.18 kips

0.220

Moment

min

-1,036.58 kip-feet

0.198

max

3,111.78 kip-feet

0.220

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Figure 3.22 Axial-force time histories for element 21.

File

FteodNeWalTest 1 ieo

I O Display extreme values and their times

(®) Plot force/moment vs. time Element Index [21

▼ | Mode: [Axial

Plot animated ^ || 3

|^J| B Loop Mode: |Axial

Elem = 21 Axial Force

Y »

-325.0

- 332 - 79 «f.o

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Figure 3.25 Shear-force time histories for element 20.

FEO Element Output

HexibleWaBTest 1 ieo

® Plot force/moment vs. time Element Index [20 Mode: [shear

Plot animated M ► II N O-OOCOO

a Elem = 20 Shear Force

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Figure 3.26 Moment time histories for element 31.

File

RexiWeWalTest 1 ieo

I O Display extreme values and their times

(®) Plot force/moment vs. time Element Index [31

▼ | Mode: |Moment

Plot animated ^ || ^ M :

^10 Loop Mode: |A»al

Y »

force (

Figure 3.27 Moment time histories for element 21.

FEO Element Output

HexibleWaBTest 1 ieo

® Plot force/moment vs. time Element Index [21 Mode: | Moment ▼ |

O Plot animated [HHHEUKHIH] l 0 00000

a Elem = 21 Moment

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Figure 3.28 Wall axial forces at 0.2 sec.

FEO Element Output

ReodWeWaBTest 1 .feo

® Plot animated fMHIfFIfJlIfMlfWI It 1 - 20000 &l Bl ^ Mode: (*

3C C.332.8

Figure 3.29 Wall shear forces at 0.220 sec.

FEO Element Output

FtedbleWaSTest 1 feo

] Mode: [Moment ▼]

~ g| [3 Loop Mode: [shear

Shear Force (kips)

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Figure 3.30 Wall moments at 0.220 sec.

Figure 3.31 Table of pile group response maximum displacements.

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Figure 3.32 Response forces for the pile groups at time 0.2200 sec.

Figure 3.33 Response forces for the pile groups at time 0.2800 sec.

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Figure 3.34 Pile group response for the pile group at node 2 and at time 0.298 sec.

Figures 3.32 and 3.33 show the instantaneous response forces at time 0.22
and 0.28 sec, respectively. At time 0.22, the longitudinal and moment
forces reach their peak values of 438.186 kips and -98.047 kip-ft for node
44, which was created at the center of rotation for the pile group. At time
0.28, the peak transverse response force is 638.156 kips at node 44.

Figure 3.34 shows the response of the pile group at node 44 at time
0.28 sec as a force-versus-displacement plot, and as force and
displacement time histories. This is the time when the displacement for
the pile group at node 44 is at its maximum location in the transverse
direction.

The Case for Dynamic Analysis:

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Node 44 (as discussed above) has a maximum transverse force of 638.156
kips, which occurs at 0.28 sec. This is the pile group node between two
flexible walls, so its transverse displacement has been affected by the
inertial effects of the walls and the over structure, as well as rotational
response. The total pile group response for the section of flexible wall
structure (with 3 pile group nodes) at this time (0.28 sec) is 662.269 kips,
this total response force is greater than the peak input force. It is also the
peak overall response. Table 3.6 shows the values generated by this data
collection. Figure 3.35 takes this information a step further by displaying
the time histories for the three forces.

Table 3.6 Transverse forces with respect to time.

Time (s)

Node 44 Force (kips)

Impact Load Force (kips)

Total Pile Group
Response (kips)

0.200

492.3062

516.400

483.616

0.280

638.1564

446.358

662.269

Figure 3.35 Time-history plot of transverse input forces, total force response for all the pile
groups, and an individual pile group response forces.

Table 3.6 and Figure 3.35 show that the overall pile group response is
dynamic because it does not track with the input force. Instead, the pile
groups respond to the input force over time due to inertial effects. Because
of this, the overall pile group response can have higher peak forces. For

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example, at the time of peak response of 0.2 sec a maximum impact force
of 516.4 kips is applied to the approach wall. The total transverse force
response of the 3 nodes representing the pile groups at the beam supports
is 483.616 kips. This 6.3% smaller summed force response is due to the
contribution of the first two terms of the equation of motion (also seen in
Appendix A) for the dynamic structural response:

[Af]{ii(f)} + [C]{li (t)} + [*]{u(t)} = {F(t)} (2.1 bis)

The summed transverse effects are smaller than the peak load because the
mass of the beams are accelerated slowly. At time 0.28 sec, when a peak
response force is recorded at node 44 for an individual pile group, the
contribution of the first two terms of the equation of motion for the
dynamic structural response is even larger resulting in an overall response
force of 662.269 kips. This is nearly 28.3% greater than the peak input
force of 516.4 kips. These observations demonstrate the importance of
applying the equation of motion for calculating pile group structural
response forces (and displacements). These differences explain why a
dynamic analysis is required versus a static analysis in which the user-
provided impact load is applied as a single peak value (e.g., determined to
be the input peak force from the time history).

For simply supported beams, the effect of load sharing between pile
groups is altered by the modal characteristics of the system (Ebeling et al.
2012). Because this system is very stiff, the inertia of the beam and pile cap
superstructure cause localized response at the node closest to the impact.
This is shown by the fact that the peak response force at node 44 is
638.156 kips, which is within 4% of the total transverse response force
(662.269 kips). Notice that the peak transverse response force at node 44
is greater than the peak transverse input force. This implies that dynamic
analysis using impulse momentum principles should be performed to
determine the greatest forces acting at any pile group.

Validation Using Impulse Calculations:

The higher peak values of the overall pile group response seem out of place
until an impulse calculation (taking the area beneath the time history
curves for input load and overall pile group response in Figure 3.35) is
performed. Despite inertial effects, the impulse of the input load must be
equivalent to the overall pile group response, if the piles do not fail. When

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an impulse calculation is performed for the overall pile group response,
the result is 463 kip-sec. The impulse for the input force is 463 kip-sec.

The difference is minimal, less than 1 %.

Load Sharing:

For this type of flexible approach wall structural system, there can be load
sharing (depending upon the structural detailing) in the longitudinal
direction starting with the first pile group beyond the point of impact.
There will also be load sharing in the transverse direction among the pair
of pile bents supporting the impact beam for an impact anywhere along
the simply supported beam. However, this structural configuration does
not have the advantage of the significant load sharing among pile groups
that the Lock and Dam 3 impact deck configuration possesses. This is
exemplified by the observation that the node 44 maximum transverse
force of 638.156 kips, which occurs at 0.28 sec, is greater than the peak
input force of 516.4 kips occurring at 0.2 sec. The pile group total
transverse response force is greater than the peak input force. For Lock
and Dam 3, the peak transverse force for the pile group possessing the
maximum peak force of any of the 96 pile groups was 56.4652 kips. The
Lock and Dam 3 dynamic structural response analysis was subjected to the
same input impact force time history specified in this analysis.

3.9 Final Remarks

In this section, the McAlpine flexible wall physical model was presented and
the mathematical model to calculate the dynamic response was also
developed. Impact_Deck is a computer program that was used to calculate
the dynamic response of an elastic beam supported over linear elastic or
plastic spring supports. The mathematical formulation modeled the ends of
the simply supported beams with no moment transfer. The impact normal
and parallel concentrated external load was located at a specified location or
assumed to have motion at a specified constant velocity. The damping effect
was considered by means of the Rayleigh damping model which depended
on the natural frequencies of the system. These natural frequencies were
calculated in an approximate way by using the linear stiffness of the pile
groups and the mass of the impact beams. The results of Impact_Deck
proved to be valid when compared to the results obtained with SAP2000.
Finally, an example was presented to show the plastic behavior of the
springs and how this result compared to the linear elastic response.

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4 Traditional Impact Beam Guard Walls

4.1 Introduction

This chapter summarizes an engineering methodology using the
Impact_Deck software for performing a dynamic structural response
analysis of a flexible impact beam supported over pile groups and subjected
to a barge impact event. For this example, a traditional model of impact
beams simply supported on bents with in-line pile groups is examined.

This example models a glancing blow impact event of a barge train
impacting an approach wall as it aligns itself with a lock is an event of
short duration; the contact time between the impact corner of the barge
train and the approach wall can be as short as a second or as long as
several seconds. The next generation of Corps approach walls is more
flexible than the massive, stiff-to-rigid structures constructed in the past in
order to reduce construction costs as well as to reduce damage to barges
during glancing blow impacts with lock approach walls. A flexible
approach wall or flexible approach wall system is one in which the wall has
the capacity to absorb impact energy by deflecting or “flexing” during
impact, thereby affecting the dynamic impact forces that develop during
the impact event.

4.2 Guard Walls - Physical Model

Guard walls are a kind of flexible wall commonly used at locks by the
Corps. Each segment of a flexible guard wall structure consists of a
continuous elastic concrete beam with a span of approximately 50 or 60 ft
long, each segment. The continuity of the beam is achieved by means of
shear key at each pile group support. This means the beams transfer the
longitudinal and transverse forces with no moment transfer at each pile
supports. The axial and transverse forces at the end of the beams are
transferred to the pile cap by means of a shear key. The shear key is a
concrete block behind the end and start of two consecutive flexible impact
beams. The length of the shear key is equal to the width of the pile cap of
the pile group. The shear key is part of the massive pile cap that rest over
the pile group. The pile group consists of two aligned piles, each with a
diameter of 5 ft 8 in. The two piles are arranged in such a way that no
torsion transfers to the pile group. A plan view drawing and a cross-section
view are presented in Figure 4.1.

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Figure 4.1 Guard wall schematic drawing.

4.3 Guard wall - Mathematical Model

The guard wall can be considered as a beam element because its length is
much greater than the other two directions. The length of each segment is
around 50 or 60 ft, heights and widths of 6 ft 7 in., and 9 ft, respectively,
are not untypical. The mathematical model is described in Figure 4.2. The
model in the analysis has two consecutive beams with longitudinal and
transverse elastic-plastic spring supports at the start, mid span, and end of
the system. These three sets of springs model the pile group at the start,
mid span, and end of the two consecutive beams. The nodes that connect
the beams to the pile cap do not transfer moment between the beams and
the center pile cap. The center pile group is modeled with three springs
(two translational and one rotational) in the generalized Impact_Deck
software formulation. This model is similar to the McAlpine alternative
flexible approach wall model but with two differences. First, no rigid link
is used in the guard wall model. Second, no rotational elastic-plastic
rotational spring is included.

The mathematical model can be done using 3-D beam elements. A 3-D
beam element has 6 degrees of freedom per node, producing 12 degrees of
freedom per element. The degrees of freedom per node are 3 translations
and 3 rotations as shown in Figure 4.3. The applied normal force F x (t) is
the impact-force time history developed using the PC-based software
Impact_Force. The applied parallel force F y (t) is a fraction of the normal
force calculated using the dynamic coefficient of friction between the barge
and the impact deck surfaces.

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Figure 4.2 Guard flexible approach wall mathematical model.

Figure 4.3 (a) Typical 3-D segment of the Impact Deck beam element, (b) Impact force applied
to the Impact Deck, (c) Typical 3-D beam element.

If the model used to describe the beam is developed in the plane, the beam
element has 3 degrees of freedom per node and 6 degrees of freedom per
element. The degrees of freedom per node are 2 translations and 1
rotation, as shown in Figure 4.4. Based on the notation of Figure 4.3, the
force and moment conditions for node i are Fi, x = Vi, Mi, x = o, Fi, y = Fi, Mi, y
= o, Fi, z = o, and M;, z = Mi, and for node/ are Ff, x = Vf, Mf, x = o, Ff, y = Ff, Mf, y
= o, Ff, z = o, and Mf z = Mf. Basically, to transform a 3-D beam element to a
2-D (plane element), the moment about the “x” axis, the moment about
the “y” axis, and the force in the “z” directions are equal to zero.

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Figure 4.4 Typical 2-D beam element used in lmpact_Deck.

The behavior of the three piles under static lateral load was done to
determine the stiffness coefficient of the springs. To have an idea of the
magnitude of the linear translational spring stiffness, refer to the
calculation in Appendix E.

4.4 Nonlinear force-deflection relationship for the springs supports

Section 2.5 discussed how the nonlinear spring response force-displacement
backbone curve was used to model plastic deformation in the pile
substructure for dynamically loaded structures. Push-over calculations were
similarly performed for the pile layout for guard wall supports.

Figure 4.5 shows the results for a fixed-head single pile analysis from
Figure 3.21 of Ebeling et al. (2012). Notice that these curves have three
linear segments with two breakpoints. As the transverse loading at the pile
cap increases, the bending moment at the top of pile-to-bent will increase
until this moment connection yields and fixity is lost. After this occurs, the
top of pile-to-bent behaves as a pinned-head condition with no constraint
against rotation being offered within this region. Observe in the push-over
curve that the rate of deformation has increased for the same incremental
load after this hinge is formed. This results in a “softer” spring stiffness
representation in this zone of the push-over curve. The pile bent system
continues to resist the increase in lateral loading up until the level of loading
that induces a second plastic hinge (shown as the second breakpoint in
Figure 4.5). This second breakpoint is reached when the piles start to hinge
at or below the mudline. Beyond this point, the push-over curve continues
to provide the same resistance for a time until the plastic hinge rotation
capacity of the piles are exhausted at this point below the mudline (refer to
section A. 10 in Appendix A of Ebeling et al. 2012 for these this capacity
computation) and can no longer support the structure.

The resulting curve for the Saul analysis (CPGA) on the wet site for two
DIP piles is shown in the force-versus-deflection, push-over curve values
listed in Table 4.1.

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Figure 4.5 Force-displacement relations from push-over analysis of a single guard wall pile.

Displacement (inches)

Table 4.1 Primary loading curve for the transverse spring model for a bent with two vertical piles.

Force

Deflection

Notes

(kips)

(inches)

(feet)

Adapted from Appendix A of Ebeling et al.
(2012)

0.0

810.0

11.28

0.94

Pile to pile cap moment capacity reached

980.0

20.272

1.68933

Flexural plastic hinges develop in piles
below mudline

980.0

26.272

2.18933

Plastic hinge rotation

In the longitudinal direction, a push-over analysis must be performed for
the pinned-head single pile condition, since there are no other piles to
constrain the bent against rotation in the direction of the longitudinal
load. The pile bent will maintain the same relative position with the top of
the piles. The pile bent will rotate with the top of piles as the moments
increase in the piles and will continue to rotate until the piles begin to

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hinge below the mudline. In this case, the push-over analysis was
performed with COM624G as a single pile model because the piles were
vertical and a pinned-head boundary condition was imposed at the top of
the pile. Again, because two piles were used, the force acting due to
deflection was doubled.

Table 4.2 Primary loading curve for the longitudinal spring model for a bent with two vertical piles.

Force

Deflection

Notes

(kips)

(inches)

(feet)

Adapted from Appendix A of Ebeling et al.
(2012)

0.0

418.0

31.1

2.59167

Flexural plastic hinges develop in piles
below mudline

418.0

37.1

3.09167

Plastic hinge rotation

4.5 Solving for the motion of the structure

The equations of motion for a structure comprised of decks supported on
groups of piles and their end-release computations similar to the guard
wall model is given in Appendix A. Appendix I gives a discussion of
Rayleigh damping with section 1 .3 giving information specific to the guard
wall model. The numerical method to be used, either HHT-a or Wilson -0
are discussed in Appendix F and G, respectively.

4.6 Validation of lmpact_Deck Computer Program

The validation of the Impact_Deck computer program for the guard wall
model was made against the results obtained from the well-known
computer program SAP2000. The beam had a total length of 100.0 ft. In
that validation, the beam was modeled with 11 nodes and 10 beam elements.
A set of linear elastic springs were located at node 1, 6, and 11 where the pile
supports were placed. The strength of the concrete was assumed as/c =
5000 psi producing a modulus of elasticity for the concrete of E =
580393.25 ksf. The beam cross-sectional area and the beam second
moment of area (moment of inertia) were 54.668 ft 2 and 517.2 ft 4 ,
respectively. The mass per linear foot of beam was calculated as m =
0.25486 kip *sec 2 /ft. A damping factor of 0.02 or 2% of the critical damping
was used in both computer programs. The force time history was the
Winfield test # 10 (Ebeling et al. 2010), as shown in Figure 4.6 and applied
at node 6 with zero translational velocity. The tangential-force time history
was the same as the transverse but multiplied by a dynamic coefficient of

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friction of 0.5. The translational spring stiffness was constant and equal to
88,125 kip/ft. Figures 4.7 and 4.8 show the dynamic-response time histories
for node 1 and node 6 in the transverse direction. Both computer programs
presented basically the same response of the system. At node 1, some
differences in magnitudes were apparent. However, the magnitudes were
very small, which was associated to numerical approximations.

Figure 4.7 Validation of lmpact_Deck against SAP2000 - Transverse
displacement at node 1.

Guard Wall
Node 1 - X Direction

- ImpactDeck-Linear — SAP2000-Linear

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Figure 4.8 Validation of lmpact_Deck against SAP2000 - Transverse
displacement at node 6.

Guard Wall
Node 6 - X Direction

4.7 Numerical Example of the Elastic-Plastic Nonlinear Response
Using lmpact_Deck

In this section, a numerical example will be shown that demonstrates the
“activation” of the plastic behavior of the pile group springs. This can be
possible if the limit elastic displacement specified by the user was low
enough to force the springs to enter into the plastic response zone. The
input data for the Impact_Deck computer program for the guard wall were
as follows. The beam has a total length of 100.0 ft long with 11 nodes and
io beam elements. A set of linear elastic springs were located at node 1, 6,
and n where the start, mid span, and end pile supports were placed. The
strength of the concrete was assumed as fc = 5000 psi producing a
modulus of elasticity for the concrete of E = 580393.25 ksf. The beam
cross-sectional area and the beam second moment of area (moment of
inertia) were 54.668 ft 2 and 517.2 ft 4 , respectively. The mass per linear foot
of beam was calculated as m = 0.25486 kip *sec 2 / ft. A damping factor of
0.02 or 2% of the critical damping was used in both computer programs.
The force time history was the Winfield test # 10 as shown in Figure 4.6
and applied at node 6 with zero translational velocity. The tangential-force
time history was the same as the transverse but multiplied by a dynamic
coefficient of friction of 0.5. The translational springs stiffness were equal
to ki = 88,126.33 kip/ft and k 2 = 44,063.165 kip/ft with a stiffness for
unload after the elastic displacement equal to kunioad = ki. The elastic
displacement that defines the elastic and plastic zone was Seiastic = 0.003 ft.

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Figure 4.11 shows the dynamic-response time history obtained for node 12
and 12’ (i.e., centerline of center pile group). It was observed that the
permanent lateral displacement of approximately 0.0035 ft when the pile
entered the plastic behavior. Figure 4.9 shows the transverse-displacement
time history of node 1 for the elastic and plastic behavior. It was presented
that no permanent displacement was reached by the transverse spring at
node 1. For the two spring models, it remained in the elastic zone.

Figure 4.10 shows the transverse-displacement time history of node 6 for
the elastic and plastic behavior. A permanent displacement was reached by
the transverse spring at node 6. For the elastic-plastic spring at node 6,
permanent displacement of about 0.0035 ft was calculated. These
behaviors can be observed in Figure 4.11, where the plastic response was
reached (second slope in the force-displacement diagram) ending with a
permanent displacement of around 0.0035 ft- It was important to observe
the two stages where the spring load and unload in the plastic zone
occured. That happened at an approximate time of 0.1 and 0.2 sec.

4.8 lmpact_Deck GUI results

The Impact_Deck GUI was also used to run the guard wall problem in this
section. This section does not provide an engineering analysis, but gives an
idea of what information was provided so that an engineering analysis
might be made.

Figure 4.9 Dynamic transverse response of node 1.

Guard Wall
Node 1 - X Direction

0.001

0.0008

0.0006

0.0004
■g 0.0002

4 0

- 0.0002

- 0.0004

- 0.0006

- 0.0008

0 0.5 1 1.5 2 2.5 3 3.5 4

Time (sec)

— Impact_Deck-Elastic ImpactDeck-PIastic

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Figure 4.10 Dynamic transverse response of spring at node 6.

Guard Wall
Node 6 - X Direction

ImpactDeck-Elastic Impact-Deck-Plastic

Figure 4.11 Plastic force-displacement of the transverse spring at

node 6.

The model input for material properties was essentially the same as
entered in section 4.7 with a few exceptions. The spring models
(transverse and longitudinal), for each pile group was returned to the
values specified in section 4.4 and shown in Figure 4.12.

The geometry for the beams was more highly resolved, with 50 beam
elements per wall section and 51 nodes per beam. The pile group nodes
were node numbers 1, 51, and 101. In all other respects, the input models
were similar.

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Figure 4.12 ImpactJDeck GUI pile group longitudinal and transverse spring model

backbone curves.

Nodal outputs provided from the FEO analysis of a flexible wall were the
longitudinal displacement, transverse displacement, and rotational
displacement (in radians) for each node at each time step of the simulation.
A table was also provided that gives the maximum displacements
(longitudinal, transverse, and rotational) for each node and the time that
the maximum displacement occured.

Figure 4.13 shows the GUI table of maximums for the example problem in
this section. From this GUI table, it is possible to tell the time step and the
node with the maximum displacement for transverse, longitudinal, and
rotational displacements (Table 4.3).

Figures 4.14, 4.15, and 4.16 show the time histories for the displacements
at nodes 51 and 50. Because some of these displacements were very small,
and the data were stored with limited precision, some of these plots
developed jaggies.

The Impact_Deck GUI also allowed the user to visualize the entire beam in
motion using an exaggerated plot of displacements longitudinally, trans¬
verse, and rotationally. Figures 4.17, 4.18, and 4.19 show the displacements
of the wall from these animated plots at the moment where the maximum

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displacement occurred, 0.332 sec for transverse displacements, 0.41 sec for
longitudinal displacements, and 0.286 sec for rotational displacements.
These data were also subjected to the jaggies because of the precision of the
stored data. The data were scaled to fit the plot.

Figure 4.13 lmpact_Deck GUI table of maximum nodal displacements for the

guard wall.

Table 4.3 Maximum nodal displacements for the guard wall example problem.

Node number

Value

Time (seconds)

Transverse

0.7706 feet

0.332

Longitudinal

0.2409 feet

0.41

Rotational

0.0147 radians

0.286

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Figure 4.14 Transverse nodal displacement time histories for node 51.

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Figure 4.16 Rotational nodal displacement time histories for node 50.

Figure 4.17 Transverse wall displacements at 0.332 sec.

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Figure 4.18 Longitudinal wall displacements at 0.41 sec.

Figure 4.19 Rotational wall displacements at 0.286 sec.

File

GuardWallTest1_pushover.feo

O Display extreme values and their times

O Plot displacement vs. time Node Index i ~ Displacement/Rotation: |z

O' Plot animated ^ || H HI 0 28600 \^\ g] Loop UisplacemenVRotation: |z

Time = 0.28600 Rotation

100.0

70.0

Y (ft) 50 0

30.0

10.0

L_ x

-(J .0147 -0.01 -0.005 0.0 0.005 0.01 0.0

147

1 — w

s -

Uz (rad)

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100

maximum forces and moments for each element and the times that the
minimum and maximum force/moment occured

Figure 4.20 show GUI tables of extreme values for the example problem in
this section. The beginning of the axial force extremes table is shown. The
shear force extremes table and moment extremes table are available by
scrolling in the interface. From this GUI table, it is possible to tell the time
step and the element with the extreme axial force, shear force, and moment
(as shown in Table 4.4). Because the impact occurs at the midpoint of the
guard wall and doesn’t move for this example, the forces are symmetric
about the impact point (node 51).

Figure 4.20 lmpact_Deck GUI table of element minimum and maximum axial
forces for the guard wall Example.

FEO Element Output

' File

Guard WallTest Ifeo

(®> Display extreme values and their times

O Plot force/moment vs. time Elementlndex i

Plot animated \i ► || N — —

e: [ Axial

[71 Looo Mode: Axial

Axial

Minimum

Maximum

Elem

Force

Time

Force

Time

; id

(kip)

(sec)

(kip)

(sec)

-138.55

0.174

138.55

0.174

-137.99

0.174

137.99

0.174

-137.43

0.174

137.43

0.174

-136.87

0.174

136.87

0.174

-136.31

0.174

136.31

0.174

-135.75

0.174

135.75

0.174

-135.18

0.174

135.18

0.174

-134.62

0.174

134.62

0.174

-134.05

0.174

134.05

0.174

-133.49

0.174

133.49

0.174

-132.92

0.174

132.92

0.174

-132.36

0.174

132.36

0.174

-131.79

0.174

131.79

0.174

-131.22

0.174

131.22

0.174

-130.65

0.174

130.65

0.174

-130.09

0.174

130.09

0.174

-129.52

0.174

129.52

0.174

-128.95

0.174

128.95

0.174

-128.38

0.174

128.38

0.174

-127.81

0.174

127.81

0.174

-127.23

0.174

127.23

0.174

-126.66

0.174

126.66

0.174

-126.09

0.174

126.09

0.174

-125.52

0.174

125.52

0.174

-124.95

0.174

124.95

0.174

-124.37

0.174

124.37

0.174

-123.8

0.174

123.8

0.174

" '■' T '

- """ '»'»

Table 4.4 Extreme Forces/Moments for the Impact Deck Example Problem.

Element number

Value

Time (seconds)

Axial

Min.

1&100

-153.99 kips

0.410

Max.

1&100

153.99 kips

0.410

Shear

Min.

-214.86 kips

0.714

Max.

214.86 kips

0.714

Moment

Min.

27&75

-2532.67 kip-
feet

0.716

Max.

26&74

2532.67 kip-feet

0.716

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101

Figures 4.21 - 4.26 show the time histories for the axial force, shear force,
and moments for the elements 50 and 51, respectively. This shows the
symmetry of the solution.

Figure 4.21 Axial force time histories for element 51.

FEO Element Output

I File

13uandWallTest1_pushover.feo

(§) Plot force/moment vs. time Element Index [ 51 ▼) Mode: | Anal

Elem

Mode: | Axial |

= 51 Axial Force

Figure 4.22 Axial force time histories for element 50.

SuandWallTest 1 _pushover.feo

<§> Plot force/moment vs. time Element Index [ 50 ▼) Mode: | Axial

Plot animated ft M Ml M M ---

Axial Force

force (kips)

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102

Figure 4.23 Shear force time histories for element 51.

Figure 4.24 Shear force time histories for element 50.

FEO Element Output
File

3uard Wall Test 1 _pushover.feo

1 Mode: Ishev

Shear Force

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103

Figure 4.25 Moment time histories for element 51.

Figure 4.26 Moment time histories for element 50.

Figures 4.27 - 4.29 show the axial force, shear force, and moment at
0.410, 0.714, and at 0.716 sec, respectively, for the entire wall. These are
created to view the animated forces in the Impact_Beam GUI.

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104

Figure 4.27 Wall axial forces at 0.410 sec.

Figure 4.28 Wall shear forces at 0.714 sec.

FEO Element Output

GuardWallTest 1 _pushoverfeo

« Plotanimated [KHTFlMlMlItWl l°™» l»t BUw M ° d * H"-I

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105

Figure 4.29 Wall moments at 0.716 sec.

Of primary importance for doing a pile-group founded flexible wall analysis
is being able to see exactly how much force each pile group will be able to
resist during an impact event. This is measured by finding the resisting
force from the spring model used for the pile group for the pile group’s
displacement. Figure 4.30 shows the table of forces/moments resisted at the
nodes representing a pile group in the Impact_Deck example model.

Figure 4.30 Table of pile group response maximum displacements.

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106

Figure 4.31 Forces at the three pile group nodes at time 0.332 sec.

Figure 4.31 displays the values for the forces and displacements for the pile
group at node 51 at time 0.332 sec. Figure 4.32 shows the response of the
pile group at node 51 at time 0.332 sec as a force-versus-displacement
plot, and as force and displacement time histories. This is the time when
the displacement for the pile group at node 51 is at its maximum location
in the transverse direction.

Figure 4.32 Transverse pile group response for the pile group at node 51 and at

time 0.332 sec.

^PM^Grou^Respon^ ^

\ File

I GuardWallTest1_pushover.feo

I © Display extreme values and their times

I © Display Spring Forces at Time: [0 3320 ;^j

O Plot Animated [Hf|[RQ[^][||][Ml[Hf| [0.33200 fej| [g] Loop Pile Group Node Index: [51 -| Degree of Freedom: [x ^1

Time = 0.33200 Node: 50 X-Axis

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107

The Case for Dynamic Analysis:

This example problem demonstrates the necessity of performing a dynamic
analysis of these pile founded walls. This example uses an impact time
history that is the result of the Winfield Test #10 (Ebeling et al. 2010). This
impact time history is applied transverse to the approach wall at a position
starting at 50.0 ft along this section of wall (at the longitudinal location of
node 51) and moved at o ft /sec along the approach wall. The peak force for
the impact time history was 516.4 kips at time 0.200 sec. Table 4.5 provides
a summary of transverse forces and the times at which they occur as well as
the magnitude of the impact force occurring at this same point in time. Peak
forces at the three times of interest are shown in bold in this table.

Table 4.5 Transverse forces with respect to time.

Time (s)

Node 51 Force (kips)

Impact Load Force (kips)

Total Pile Group
Response (kips)

0.200

398.9343

516.400

232.6187

0.332

663.9989

320.2922

816.9583

0.392

606.7369

174.0544

934.5545

Node 51 (as discussed above) has a maximum transverse force of 663.999
kips, which occurs at 0.332 sec. This is the pile group node between two
flexible approach wall impact beams, so its transverse displacement has
been affected by the inertial effects of the walls and the over structure, as
well as rotational response. The total pile group response for the section of
flexible approach wall structural model (with 3 pile group nodes) at this
time (0.332 sec) is 816.958 kips.

The peak overall force occurs at a later time for the guard wall bents (at
0.392 sec) than in the input force time history (0.2 sec). This maybe due
to the natural frequency of the system with less stiff supporting piles but
more analysis (e.g., considering modal analysis of the piles as discussed in
Appendix E of Ebeling et al. 2012) will be required. The peak overall force
occurs at 0.392 sec and have a value of 934.555 kips, exceeding the peak
input force of 516.4 kips by a good margin.

Figure 4.33 takes this information a step further by displaying the time
histories for the three forces.

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108

Figure 4.33 Time-history plot of transverse input forces, total force response for all the pile
groups, and an individual pile group response forces.

Time Histories of Transverse Load and Response
for Guard Wall Model

Input Load

Overall Pile Group Response

Maximum Individual Pile Group
Response (Node 51)

From Table 4.5 and Figure 4.33, it can be seen that the overall pile group
response is dynamic because it does not track with the input force.

Instead, the pile groups respond to the input force over time due to inertial
effects. Because of this, the overall pile group response can have higher
peak forces. For example, at the time of peak response of 0.2 sec a
maximum impact force of 516.4 kips is applied to the approach wall. The
total transverse force response of the three nodes representing the pile
groups at the beam supports is 232.619 kips. This 55% smaller summed
force response is due to the contribution of the first two terms of the
equation of motion (Equation 2.1, also seen in Appendix A) for the
dynamic structural response:

M{ fi ( f )Wc]H f )} + MM0M F (0} (2-1 bis)

The summed transverse effects are thought to be smaller than the peak
input force because the mass of the beams is accelerated slowly. At time
0.332 sec, when a peak response force is recorded at node 51 for an
individual pile group, the contribution of the first two terms of the equation

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109

of motion for the dynamic structural response is even larger, resulting in an
overall response force of 663.999 kips. This is nearly 28.6% greater than the
peak input force of 516.4 kips (occurring at 0.2 sec). Additionally, the
overall response does not reach a peak until 0.392 sec with an even larger
value of 934.555 kips. This greater overall force results from the pile groups
all developing a positive deformation at the same time. These observations
demonstrate the importance of applying the equation of motion for
calculating pile group structural response forces (and displacements). These
differences explain why a dynamic analysis is required versus a static
analysis in which the user-provided impact load is applied as a single peak
value (e.g., determined to be the input peak force from the time history).

For this type of simply supported impact beam structural system with two
beams, the modal contribution characteristics of the substructure system
(Ebeling et al. 2012) may be important. Because this pile substructure
system is relatively flexible, the inertia of the impact beams and pile cap
superstructure may cause vibrations to be more in sync with the impact
event. Expansion of the current dynamic Impact_Deck model would be
required to account for this feature.

Validation Using Impulse Calculations:

The higher peak values of the overall pile group response seem out of place
until an impulse calculation (taking the area beneath the time history
curves for input load and overall pile group response in Figure 3.35) is
performed. Despite inertial effects, the impulse of the input load must be
equivalent to the overall pile group response, if the piles do not fail. When
an impulse calculation is performed for the overall pile group response,
the result is 461 kip-sec. The impulse for the input force is 463 kip-sec.

The difference is minimal, less than 1 %.

Load Sharing:

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110

that the Lock and Dam 3 impact deck configuration possesses. This is
exemplified by the observation that the node 51 maximum transverse force
of 663.999 kips, which occurs at 0.392 sec, is greater than the peak input
force of 516.4 kips occurring at 0.2 sec. The pile group total transverse
response force is greater than the peak input force. For Lock and Dam 3,
the peak transverse force for the pile group possessing the maximum peak
force of any of the 96 pile groups, was 56.4652 kips. The Lock and Dam 3
dynamic structural response analysis was subjected to the same input
impact-force time history specified in this analysis.

4.9 Final Remarks

In this section, the flexible guard wall physical model was presented and
the mathematical model to calculate the dynamic response was also
developed. Impact_Deck calculated the dynamic response of an elastic
beam supported over linear elastic or plastic spring supports. The
mathematical formulation also modeled the center pile group connection
to the ends of the simply supported impact beams with zero moment
transfer. The impact normal and parallel concentrated external load can
be located at a specified location or can be assumed to have motion at a
specified constant velocity. The damping effect was considered by means
of the Rayleigh damping model which depended on the natural
frequencies of the system. These natural frequencies were calculated in an
approximate way by using the linear stiffness of the pile groups and the
mass of the impact beams. The results of Impact_Deck proved to be valid
when compared to the results obtained with SAP2000. Finally, an example
was presented to show the plastic behavior of the springs and how this
result compared to the linear elastic response.

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111

5 lmpact_Deck Graphical User Interface
(GUI)

5.1 Introduction

This section introduces the Impact_Deck GUI, which provides pre¬
processing and post-processing capabilities to the Impact_Deck engineering
code. The purpose of the GUI is to provide the user a way to specify the
input model types, input model parameters, analyze the input, and visualize
the output.

The program has a simple menu that allows the user to create a new data
input set, open an existing set of input data, save the current input data in
an existing or new input file, and exit the program. On the line below the
menu, enter the title of the project; this will provide a reference for the
user.

Beneath the title bar, the Impact_Deck GUI uses a tabbed data input
scheme where input data were grouped by related data and functionality.
The Introduction Tab shows cross-section and plan views of examples of
the different types of structures that can be analyzed with the
Impact_Deck software (Figure 5.1).

Figure 5.1 Introducing lmpact_Deck.

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112

The following sections discuss the other tabs in the Impact_Deck program.

5.2 Geometry Tab

The Geometry Tab is made up of subsections of data that specify the
positions and velocities associated with the wall model and the source of
the impact (typically, a barge train). These positions and velocities are
entered in feet and feet/second, respectively. The rest of the program
assumes English units for version l.o. The geometry information assumes
a right-handed coordinate system, with the lock approach wall lying along
the Y-axis (i.e., the longitudinal direction) and the X-axis proceeds into the
wall (i.e., the transverse direction).

The first section, at the upper left corner of the tab, is the selection of the
type of wall to analyze. The choices reflect the three types of walls
discussed previously; flexible approach walls, guard walls, and impact
decks. Selecting any of these options changes the inputs available for the
rest of the program. Figures 5.2 through 5.4 show the Geometry Tab when
each of the options is selected.

Figure 5.2 Geometry for a flexible wall.

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113

Figure 5.3 Geometry for a guard wall.

Figure 5.4 Geometry for an lmpact_Deck.

The second subsection is unaffected by the approach wall type. This is the
load information subsection. In this subsection are the inputs for the Y-
position for the start of the impact and the velocity at which the impact

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114

travels along the wall. The impact time history, which is input on the next
tab, determines the duration of the analysis.

For flexible approach walls and guard walls, only two simply supported
impact beams on three pile groups are included in the model. The
Additional Pile Groups section allows the user to specify the number of
pile groups that exist prior to and after the modeled section, so that those
pile groups can contribute to the response.

An impact deck consists of a series of impact monoliths with each
monolith supported by clustered groups of piles. The impact deck
structure is modeled as an entire set of beams spanning between multiple
pile supports. Each pile support is modeled as a pair of transverse and
longitudinal nonlinear springs (established through a push-over analysis
as outlined in Ebeling et al. 2012). The fixity for the Starting End
subsection allows the user to specify how the starting end of the approach
wall, at the end away from the lock, is affixed. The fixed support constrains
the end point in translation and rotation. The simple support has the end
of the wall section resting on the connection to the end cell.

At the right of the tab is the nodal input section for the wall. Because the
wall always proceeds along the Y-axis, all of the nodes specified by the user
can be entered with only a Y-coordinate. This approach wall has its origin
at the point along the approach wall that is furthest from the lock
chamber.

For the flexible approach wall and guard wall models, the first and last
nodes in the wall are automatically assigned to be pile group nodes, but
the user must flag one of the internal nodes as associated with the central
pile group. The pile group nodes are the nodes where the spring models for
the piles resist the impact on the wall. The process for creating the central
pile group will be discussed further in this section.

For the Impact_Deck model, most of the nodes are connected to a pile
group, but some nodes represent only the connectivity between the
sections of impact deck monoliths. These nodes are called inter-monolith
nodes. The inter-monolith nodes are not connected to a pile group and
have different end-release properties. The process for creating the inter¬
monolith nodes is similar to the method used to create the central pile
group for the flexible wall and guard wall models.

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115

Before discussing the creation and deletion of nodes, the visualization of
nodes must be discussed. In the node input section to the right of the tab,
there is a list containing the node information. The list will provide the
position of the node and if the node is a central pile group/inter-monolith
node or not. This list is given to provide the user with specific nodal data.

At the bottom of the tab is the Input Plot area. This plot shows the existing
nodes, their connectivity, load conditions on the wall, and the number of
pile groups prior and after the displayed wall segment (for flexible wall
and guard wall models). The position of the starting point of the barge
train impact is shown with an arrow pointing at the wall. The direction of
the barge train velocity (parallel to the approach wall) is input prior to
specifying an impact-force time history. After an impact-force time history
has been selected, the time history is displayed from the starting point
until the end of the time history due to the velocity of the impact.

Nodes are displayed with different colors depending on whether the pile is a
pile group/inter-monolith node. Blue nodes represent unsupported nodes
(no pile group) for the flexible approach wall and guard wall models and
regular pile group nodes for the Impact_Deck model. The red nodes
represent the pile group nodes for the flexible approach wall and guard wall
models and inter-monolith connection nodes for the Impact_Deck model.

When the mouse is moved across the Input Plot window with no mouse
button pressed, the node that the cursor is closest to will be highlighted,
and information about that node will be presented to the right of the Input
Plot window. This is shown in Figure 5.3 and Figure 5.4.

The view in the Input Plot window can be zoomed by click-dragging with
the right mouse button (Figure 5.5). The view will be changed to display
everything in the selected region with the aspect ratio maintained
(Figure 5.6). The button to the right of the Input Plot window with a global
map on it is the Zoom Extents button. Clicking this button reveals the entire
wall as it is currently defined.

Nodes can be selected by click-dragging with the left mouse button. Nodes
that are surrounded by the dragged bounding box will all be selected.
Multiple selection regions are not permitted at this time. Selected nodes are
drawn with a line through them to differentiate them from the unselected
nodes (Figure 5.7). Selected nodes maybe copied or deleted, as discussed
below.

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116

Figure 5.5 Zooming in the input plot section.

Input Plot

o Pile Group Node

o Beam Node

Node 68

r Y

YPos:

415.9791666

Figure 5.6 The zoomed view.

Input Plot

Node 86

YPos:

524.1666665

r Y

Figure 5.7 Selected nodes are highlighted.

Input Rot

o i ^

■ 14 o

Node 86

- -- --- ° PTTtTttTtTTi

r Y

' • T 0

YPos:

524.1666665

There are three ways to create nodes in the node list. These three methods
allow the user to input individual nodes, multiple nodes using
interpolation, and copying and pasting nodes.

The Single Node Input subsection allows the user to specify a location
along the wall as a Y-axis position. Recall that the wall lies along the Y-axis
and that the X-axis is into the wall with a right-handed coordinate system.
A checkbox permits the user to specify whether this node is a central pile
group node for the flexible wall and guard wall models and an inter¬
monolith node for the Impact_Deck model. Clicking the Add Node button
in this subsection adds the node to the node list and plots it in the Input
Plot area. Adding a node at the location where a node already exists will
not create a new node, but can change the status of the node to or from a
central pile group/inter-monolith node.

Multiple nodes can be input using the Interpolated Node Input subsection.
When a start and end position are entered with a number of divisions
between nodes in that distance, nodes will be linearly distributed in that
distance. The number of nodes placed will be equal to the number of
divisions plus one; a node is placed at the start position and then the
following nodes are placed at the total length divided by the number
divisions away from the previous node, until the end point is reached.

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117

Because there can be only one central pile group, interpolated nodes are
not allowed to set that status. However, each node could be an inter¬
monolith node, so setting that status is allowed. Again, nodes that will be
placed at the same location as existing nodes will not create a new node,
but can change the status of the existing node.

If there are nodes selected, then nodes can be created by clicking the Copy
Selected Nodes button. When the button is selected, the Copy Selected
Nodes dialog will appear that asks for an offset for the selected nodes
(Figure 5.8). Clicking the Cancel button will terminate the copy event, but
clicking Accept will cause the Copy Selected Nodes with Offset dialog box
to open (Figure 5.9). Clicking the Accept button in this dialog creates a
copy of the selected nodes with their attributes at the offset location
relative to the original nodal positions (Figure 5.10). The original nodes
are then deselected so they will not be copied again, because the Copy
Selected Nodes with an Offset dialog box stays open in case multiple
copies of the nodes needs to be made at the same relative distance.

Clicking the Cancel button does not copy the last set of selected nodes, and
terminates the operation.

There are two methods for removing nodes. The Empty Node List button
removes every node in the model. The Delete Selected Nodes button
removes only the selected nodes, as shown in the Input Plot window.

Figure 5.8 Entering an offset to copy
selected nodes.

Figure 5.9 Confirming the offset
copy (which can be performed
multiple times).

Copy Selected Nodes with Offset

Copy selected nodes with offset of 1000?

OK || I Cancel

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118

Figure 5.10 Selected nodes are copied at the offset position.

5.3 Impact Time History Tab

The Impact Time History Tab is very simple. There is a method to browse
for a time-history file and there is a button to adjust the time history to
allow time for the pile-founded impact deck or flexible approach wall to
reach a state where little-to-no displacements occur.

The Browse button permits the user to bring in an impact time history file.
Currently, the only format supported is the “.ETH” format output by
Impact_Force. When an impact time history has been chosen the path for
the file selected, relevant comments about the time history and actual
values are displayed per Figure 5.11.

The time history can be altered to add samples of zero force at the end to
allow time for the dynamic structural response to settle after the
deformations have completed. Clicking the Extend Force Time History
button brings up the Extend Force Time History dialog. The user can
specify a new length for the time history and the time history will be
extended to the new length (Figure 5.12).

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Figure 5.11 Input for an impact time history.

Figure 5.12 Extending a time history with
0.0 value.

5.4 Beam Properties Tab

The beam properties are different depending on whether a flexible
approach wall model is chosen or if the guard wall or Impact_Deck model
is chosen. For the flexible approach wall model, the beam properties also
include the ability of the beam to flex and rotate differently than the guard
wall beam or impact deck’s deck (Figures 5.13 and 5.14).

The primary beam input, which is shared between the three models, are
the Modulus of Elasticity, Cross-sectional Area, Moment of Inertia, Mass
per Unit Length, Damping Ratio, and the Coefficient of Friction Between
the Barge and the Beam/Deck.

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120

Figure 5.13 Beam properties tab as it appears for a flexible wall.

a? Impact_Deck FlexibleWallTestl.idf

File

™e: FLEXIBLE WALL TEST

Introduction Geometry Impact Time History [ Beam Properties] Pile Ouster Springs Analyze Output

Beam Properties

Modulus of Elasticity: 580393.25
Cross-sectional Area: 54.6675

Moment of Inertia: 517.1967

Mass Per Unit Length: 0.25466

Damping Ratio: 0.1

Coefficient of Friction Between Barge and Deck: 0.5

Beam Properties (Longitudinal)

Modulus of Elasticity
Cross-sectional Area
Moment of Inertia
Mass Per Unit Length

Beam Properties (Transverse)

Modulus of Elasticity
Cross-sectional Area
Moment of Inertia
Mass Per Unit Length

Length: 7.708

kips/fT2

fT2

fT4

(kips*sec"2)/ft

kips/fT2

fT2

fT4

(kips*sec~2)/ft

kips/fT2

fT2

fT4

(kips*sec"2)/ft

Figure 5.14 Beam properties tab as it appears for an impact deck or guard wall.

Tit'e: IMPACT DECK TEST PROBLEM 1

Introduction Geometry Impact Time History [ Beam Properties! Pile Ouster Springs Analyze Output

Beam Properties

Modulus of Elasticity
Cross-sectional Area
Moment of Inertia
Mass Per Unit Length
Damping Ratio

kips/fT2

fT2

fT4

(kips*sec~2)/ft

Coefficient of Friction Between Barge and Deck: 0.5

Beam Properties (Longitudinal)

Modulus of Elasticity:
Cross-sectional Area:

Moment of Inertia:
Mass Per Unit Length:

Beam Properties (Transverse)

Modulus of Elasticity:
Cross-sectional Area:

Moment of Inertia:
Mass Per Unit Length:

Length: [(T

kips/fT2

ft~2

fT4

(kips*sec~2)/ft

kips/fT2

fT2

fT4

(kips’sec^yft

ERDC/ITL TR-16-1

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Flexible walls have input for properties acting along the beam and
transverse to the beam. The longitudinal properties are the Modulus of
Elasticity, Cross-sectional Area, Moment of Inertia, and Mass per Unit
Length for the length of the wall. The transverse properties are the
Modulus of Elasticity, Cross-sectional Area, Moment of Inertia, Mass per
Unit Length, and the Length of the wall beam.

5.5 Pile Cluster Spring Tab

For the Impact_Deck software, pile clusters are modeled as non-linear
springs. In the Pile Cluster Spring Tab, these non-linear pile cluster
springs are defined.

Again, there is a difference based on the wall model being used. Because
the flexible approach wall model has pile groups that can rotate, entry of a
rotational spring model is allowed (Figure 5.15). Figure 3.1 shows an
example of a clustered three-pile group providing rotational resistance,
with Appendix E summarizing the computations leading to the value
assigned to its rotational spring stiffness. Because the guard wall and the
impact deck wall models do not allow the pile caps to rotate, rotational
springs are not defined (Figure 5.16).

Figure 5.15 Beam properties tab as it appears for a flexible wall.

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Figure 5.16 Beam properties tab as it appears for an lmpact_Deck.

To the right of this tab, there is a graph of the currently defined springs
(longitudinal, transverse, and possibly rotational). Radio buttons next to
these graphs allow the user to choose the spring definition currently being
edited. The currently edited spring definition has a graph with a white
background, and its values are displayed in the Edit subsection to the left.

In the Edit subsection, there are three changes the user can apply to the
current spring

• Change the Unload/Reload characteristics of the curve

• Add a force-displacement point to the curve

• Delete a force displacement point from the curve

There are two values that can be entered to adjust the Unload/Reload
characteristics of the spring model. The first value is the amount of elastic
lateral displacement allowed before plastic deformation occurs. The
second value is the unload/reload stiffness, which is the force-versus-
displacement slope that will be followed from the peak displacement and
its force after a plastic deformation has occurred. These values will
typically be the point where the first point on the curve is and the slope the

ERDC/ITL TR-16-1

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first linear segment makes on the curve, but entry of these values lends
more flexibility.

Adding a point to the curve requires that the user enter a displacement and
the force at that displacement, and then press the Add Point to List button.
The points in the list (which are shown below the Point Entry subsection)
are sorted by displacement. The points in the list are plotted in the current
graph. Multiple forces may not be entered at the same displacement. When
a new displacement/force pair is at the same displacement as an existing
point, the new point overrides the previous point.

To delete a point in the current spring model, the user can click on the row
in the list for the point. That point is highlighted in blue. Pressing the
Delete Point from List button removes the point from the list.

These operations work for each of the spring model types.

5.6 Analyze Tab

The Analyze Tab (Figure 5.17) allows the user to:

• View his wall model,

• Choose a solution method,

• Set the capture rate for data,

• Select specific elements for more direct Finite Element Data, and

• Perform an analysis.

The input plot display mirrors the input plot display on the Geometry Tab,
but does not allow the user to select nodes. In this way, a wall model can
be quickly verified.

There are two solution methods that can be applied to compute the
dynamic structural response for the approach wall model: the HHT-a
method and the Wilson -0 method. Each method has a different tolerance
for the solution to be met. For the HHT-a method an alpha tolerance can
be specified that must be met to finish the simulation. Similarly, for the
Wilson -0 method, a value for Theta can be specified such that the method
will converge.

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Figure 5.17 Analyze tab input for analysis method and specified output.

Time-history response analysis of a MDOF structural system model can
result in large files of computed time histories for the various output
variables. In order to reduce the size of the output files, the following feature
has been added: each time step in the computed simulation results can be
captured or the user can skip past a designated number of time steps.
Entering l in the Capture every Time Step subsection guarantees that every
time step is captured; entering 2 in the subsection means that every other
time step is captured; and entering 3 in the subsection means that every
third time step is captured and is repeated thusly.

If the checkbox labeled Output Finite Element Data is checked, then the
user can select specific finite element nodes that he/she would like to have
text output for. The list next to the checkbox shows the nodes that will
currently be output. To change the list, press the Manage Output button.

Pressing the Manage Output button brings up a dialog box that gives the
user the power to select output nodes (Figure 5.18). A diagram similar to
the Input Plot of the Geometry tab is on the left of the Manage Output
dialog. This diagram can be zoomed in much the same manner and a
Zoom Extents button is provided. The node selection process works in a

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different fashion in the Manage Output dialog. When the user left click-
drags an area in the diagram, the nodes are toggled between being selected
and not selected. In other words, if a node was selected when the mouse
was left click-dragged over it, it will be deselected. This allows the user to
select multiple groups of nodes by selecting a group and then deselecting a
subgroup. Selected nodes are shown highlighted.

Figure 5.18 Selecting finite element nodes
where data will be captured.

A current list of selected nodes is displayed at the right of the dialog. There
are also options to delete a node by selecting it from the list and then
pressing the Remove Node button, and to clear the entire list by pressing
the Clear List button. In order to use the list chosen in the Manage Output
dialog, press the Accept button. Pressing the Cancel button keeps the
original list.

At the bottom of the Analyze Tab is the button labeled Perform Analysis.
When this button is pressed, the processor is started with the current
input data.

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5.7 Output Tab

In the Output tab, output data files can be selected by the users for
visualization of results. There are two main files that are output for each
Impact_Deck processor run. There is the .FEO (Finite Element Output) file
which contains the node and element results in a time domain solution, and
there is the Run File output in .OUT format that explains how the run
proceeded and provides only the requested node input from the Analysis
Tab, so the user does not have to look for specific information in a very large
file.

The Output tab is therefore broken into two sections (Figure 5.19). Each
section has a Load Output button, which is used to browse for the specific
file. These files are automatically populated with the output from the
current input file when the Perform Analysis button is pressed on the
Analyze tab. When a file is selected for either section, that section’s
buttons become enabled, and that file’s data is available to be visualized
(Figure 5.20).

Figure 5.19 Output tab for selecting and viewing select data.

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Figure 5.20 Output tab with selected data.

Title: IMPACT DECK TEST PROBLEM 1

Introduction Geometry Impact Time History Beam Properties Pile Ouster Springs Analyze | Output

FEO Output Options

Data Path : | lmpact_Deck_TestProbleml .feo

Load Output-

Show Nodal Information

Show Element Information

Show Pile Group Response

Run Output Options

Data Path : lmpact_Deck_TestProbleml .out

Show Run Output

For the FEO output file, there are three data to visualize: the Nodal
Information, the Element Information, and the Pile Group Response. For
the Run file output, the user can see the text file detailing how the program
ran, and the specific finite element information selected on the Analyze
Tab. These options are discussed below.

5.7.1 FEO Nodal Output

Pressing the Show Nodal Information button from the Output Tab brings
up the FEO Nodal Output window. The window has a minimal menu, with
only options to save the currently displayed information (either as text or
as a .PNG bitmap graphic file for graphical data) and to close the window.
The currently selected file name is displayed immediately below the menu.
Beneath the menu are radio buttons that allow the user to present
different information from the .FEO file.

The FEO Nodal Output window presents nodal information in three ways

1. The maximum values of all the nodes are presented as text

2 . Individual nodal properties are shown as a 2-D plot of displacement versus
time

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128

3. The entire structure is shown as a 2-D animated plot with the structure
along the Y-axis, and the displacement along the X-axis.

When the “Display extreme values and their times” radio button is selected,
a list of nodes with each node’s maximum values of displacement (longi¬
tudinal, transverse, and rotation) and the times when those maximums
occurred is displayed (Figure 5.21). The longitudinal displacements occur
along the wall beam, transverse displacements occur into the wall beam,
and rotations occur about the node in the beam.

Figure 5.21 FEO nodal output window showing maximum nodal values for all the

nodes.

When the “Plot displacement vs. time” radio button is selected, the user
can select a node to view using the “Node Index” combo box. When a node
has been selected, the user can select whether to plot the node’s X-
displacement, Y-displacement, or Rotation versus Time in the plot below
by selecting from the “Displacement/Rotation” combo box.

When these options have been selected, the graphic window below shows
the relevant plot (Figures 5.22 through 5.24). At the top of the display is the
selected node number and selected displacement. To the left of the plot is a
display of the coordinate system and the wall with nodes and connectivity.
The selected node is displayed in red to highlight where the node is in

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129

Figure 5.22 Graph of node 70 X-displacement vs time.

FEO Nodal Output

File

lmpact_Deck_T estProbleml .feo

® P '°* a " imated jMTMlM P°000~

| Displacement/Rotation: [x~

Displacement/Rotation: [x ▼]

Node =70 X Displacement

Figure 5.23 Graph of node 70 Y-displacement vs time.

FEO Nodal Output

lmpact_Deck_TestProbleml .feo

Displacement/Rotation:

[ 7 ] i_Qop Displacement/Rotation: [x =]

Node = 70

Y Displacement

disp. (ftV

ERDC/ITL TR-16-1

130

Figure 5.24 Graph of node 70 Z-displacement vs time.

FEO Nodal Output

File

lmpact_Deck_T estProbleml .feo

® fllSOTUMliMlWI P°°°0~

] Displacement/Rotation: (z

Displacement/Rotation: |x ▼]

Node = 70 Rotation

relation to the wall. The rest of the area is the 2-D plot of the displacement
versus time for the selected node and displacement. When the mouse is
placed over this plot, a tooltip is shown revealing the displacement for the
time that the cursor is over.

It should be noticed that some displacements have some artifacts in their
plots, typically shown as “staircase steps”. The “staircase steps” occur when
the resolution of the data, with small numbers, outstrips the precision of the
floating point variables used to represent the data. Aggregating this data
leads to discontinuities in the curve which appear as “steps”. These steps
occur in the low amplitude region of the curve which is inconsequential to
the design. Notice in Figure 5.24 that the image depicts the vertical
displacement of a node of the impact model and that the vertical displace¬
ment only varies by 0.0009 inches at the extremes, which will have an
inconsequential bearing on the response forces of the structure.

When the “Plot animated” radio button is selected, the user can select
whether to plot the wall’s X-displacement, Y-displacement, or Rotation
versus Time in the plot below by selecting from the “Displacement/

ERDC/ITL TR-16-1

131

Rotation” combo box. The displacements/rotations at each node are linearly
interpolated from node to node.

Immediately next to the radio button is the animation control. This control
has typical buttons that

• Return the animation to the beginning time step,

• Step to the previous time step,

• Play the animation,

• Pause the animation,

• Step to the next time step, and

• Take the animation to the last time step.

Next to the buttons in the animation control is the current time display.
The time display can be stepped up or down using the arrows next to the
display. A checkbox next to the current time display allows the user to
select if the animation will loop to the beginning or stop when the last time
step has been reached.

When these options have been selected, the graphic window below shows
the relevant plot (Figures 5.25 through 5.27). At the top of the display is
the current time and selected displacement. To the left of the plot is a
display of the coordinate system and the wall with nodes and connectivity.
The plot of the impact-load time history from the starting position of the
impact to the ending impact location, based on the velocity of the
impacting barge train, is displayed along the wall. The current location of
the load is displayed as a red line along the time history.

Because the impact can have a very low velocity, it can sometimes be hard
to see the full impact in the wall view. For that reason, the time history
display with a red line showing the current time is shown at the bottom left
of the plot for the input impact-force time history.

The rest of the area is the 2-D plot of the displacements of each node
relative to the wall. The scaled displacements are shown in the scale along
the X-dimension. The nodal displacements are connected with line
segments, effectively linearly interpolating the displacements between
nodes.

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132

Figure 5.25 Animated graph of wall X-displacement.

FEO Nodal Output

File

lmpact_Deck_T estProbleml .feo

O Plot displacement vs. time Node Index; [to -r | Displacement/Rotation: [z~

® Plot animated MHliyifllHlNl 008400 g] Loop Displacement/Rotation:

Time = 0.08400 X Displacement

0.04 0.05

0.06 0.0663

Figure 5.26 Animated graph of wall Y-displacement.

FEO Nodal Output

lmpact_Deck_TestProbleml .feo

® Plot animated IWIRTFimHIll • .08400 [gj| g] [joop Displacement/Rotation;

Time = 0.08400 Y Displacement

838.6667

800.0

0.0002 0.0005 0.0007 0.001 0.0012 0.0015 0.0017

Oy (ft)

ERDC/ITL TR-16-1

133

5.7.2 FEO Element Output

Pressing the Show Element Information button from the Output tab
brings up the FEO Element Output window. The window has a minimal
menu, with only options to save the currently displayed information
(either as text or as a .PNG bitmap graphic file for graphical data) and to
close the window. The currently selected file name is displayed
immediately below the menu. Beneath the menu are radio buttons that
allow the user to present different information from the .FEO file.

The FEO Nodal Output window presents element information in three
ways

1. The minimum and maximum values of all the elements are presented as
text

2. Individual elements are shown as a 2-D plot of force/moment versus time

3. The entire structure is shown as a 2-D animated plot with the structure
along the Y-axis, and the force/moment along the X-axis.

When the “Display extreme values and their times” radio button is
selected, a set of lists containing each element’s minimum and maximum

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134

force/moment (axial force, shear force, and moment) and the time that
those extreme values occurred is displayed (Figure 5.28).

When the “Plot force/moment vs. time” radio button is selected, the user
can select an element to view using the “Element Index” combo box. When
an element has been selected, the user can decide whether to plot the
element’s Axial force, Shear force, or Moment versus Time in the plot
below by selecting from the “Mode” combo box.

Figure 5.28 FEO element output window showing maximum and minimum force and
moments acting on all the elements.

FEO Element Output

File

lmpact_Deck_TestProblem1.feo

(«} Display extreme values and their times

1 O Plot force/moment vs. time

Element Index [

1 ▼ | Mode:

| Axial ▼]

O Plot animated [MINIMI! l[ Miff]

Loop Mode: [Axial ▼]

Axial

Axial *

Minimum

Maximum

Elem

Force

Time

Force

Time

(kip)

(sec)

(kip)

(sec)

-251.45

0.21

251.45

0.21

-251.45

0.21

251.45

0.21

-251.45

0.21

251.45

0.21

-251.46

0.21

251.46

0.21

-251.46

0.21

251.46

0.21

-251.47

0.21

251.47

0.21

-251.47

0.21

251.47

0.21

-251.48

0.21

251.48

0.21

-251.49

0.21

251.49

0.21

-251.5

0.21

251.5

0.21

-251.52

0.21

251.52

0.21

-251.53

0.21

251.53

0.21

-251.55

0.21

251.55

0.21

-251.56

0.21

251.56

0.21

-251.58

0.21

251.58

0.21

-251.6

0.21

251.6

0.21

-251.65

0.21

251.65

0.21

-251.59

0.21

251.59

0.21

-251.64

0.21

251.64

0.21

-251.66

0.21

251.66

0.21

-251.69

0.21

251.69

0.21

-251.72

0.21

251.72

0.21

-251.74

0.21

251.74

0.21

-251.77

0.21

251.77

0.21

-251.8

0.21

251.8

0.21

-251.84

0.21

251.84

0.21

-251.87

0.21

251.87

0.21

nr-, ft-.

A A1

When these options have been selected, the graphic window below shows
the relevant plot (Figures 5.29 through 5.31). At the top of the display is
the selected element number and selected force/moment. To the left of the
plot is a display of the coordinate system and the wall with nodes and
element connectivity. The selected element is displayed in red to highlight
where the element is in relation to the wall. The rest of the area is the 2-D
plot of the force/moment versus time for the selected element and
force/moment. When the mouse is placed over this plot, a tooltip is shown
revealing the force/moment for the time that the cursor is over.

ERDC/ITL TR-16-1

135

ERDC/ITL TR-16-1

136

Figure 5.31 Graph showing element 60 shear force vs time.

When the “Plot animated” radio button is selected, the user can select
whether to plot the wall’s Axial force, Shear force, or Moment versus Time
in the plot below by selecting from the “Mode” combo box. The
force/moment at each element is linearly interpolated from element to
element. The animation control behaves in the same manner as it did for
the FEO Nodal Output window.

When these options have been selected, the graphic window below shows
the relevant plot (Figures 5.32 through 5.34). At the top of the display is
the current time and selected force/moment mode. To the left of the plot is
a display of the coordinate system and the wall with nodes and element
connectivity. The plot of the impact-load time history from the starting
position of the impact to the ending impact location, based on the velocity
of the impacting barge train, is displayed along the wall. The current
location of the load is displayed as a red line along the time history.

ERDC/ITL TR-16-1

137

Figure 5.32 Animated graph of axial forces acting on the wall.

FEO Element Output

lmpact_Deck_TestProblem1 feo

"BBiw

0.0

Axial Force (kips)

Figure 5.33 Animated graph of moments acting on the wall.

FEO Element Output

| lmpact_Deck_Test Problem 1 .feo

® Plotanimatad [MfflfMIIlHlll M ® “■»

Mode: Moment ▼

0.0

Mom ent (kip-ft)

5000 £0662.18

ERDC/ITL TR-16-1

138

Figure 5.34 Animated graph of shears acting on the wall.

FEO Element Output

lmpact_Deck_TestProblem1 feo

Plot force/moment vs. time Element Index: [ 70 ▼] Mode: [Moment

® Plot animated WBSMMM [0.08400 [4-j| |7] Loop Mode: [shear

-250.0 0.0 250.0

Shear Force (kips)

The rest of the area is the 2-D plot of the force/moment of each element
relative to the wall. The scaled force/moment is shown in the scale along
the X-dimension. The element force/moment is projected to the nodes and
the values connected with line segments, effectively linearly interpolating
the force/moments between elements.

5.7.3 FEO Pile Group Response

Pressing the Show Pile Group Response button from the Output tab brings
up the Pile Group Response window. The window has a minimal menu,
with only options to save the currently displayed information (either as
text or as a .PNG bitmap graphic file for graphical data) and to close the
window. The currently selected file name is displayed immediately below
the menu. Beneath the menu are radio buttons that allow the user to
present different information from the .FEO file.

The Pile Group Response window presents element information in three
ways:

l. The minimum and maximum values of all the pile group nodes are
presented as text

ERDC/ITL TR-16-1

139

2. The force and moment values at any selected time during the analysis are
presented as text

3. The entire structure is shown as a 2-D animated plot with the structure
along the Y-axis, and the pile group force/moment along the X-axis.

When the “Display extreme values and their times” radio button is selected,
a list containing each pile group nodes’s minimum and maximum group
response force/moment (longitudinal force, transverse force, and moment)
and the time that those extreme values occurred (Figure 5.35) and peak
displacements and when they occurred (Figure 5.36) are displayed.

When the “Display Spring Forces at Time” radio button is selected, the
user can enter a time in the simulation and get a snapshot of the forces
acting at each pile group node at that time (Figure 5.37). At the base of the
list, the forces are totaled to give an overall force acting at all of the pile
group nodes. Additionally, the total impulse calculations for longitudinal
and transverse forces and the moments are presented at the end of the list.

Figure 5.35 Pile Group Response Maximum and Minimum Forces and Moments.

Pile Group Response

lmpact_Deck_TestProbleml feo

1 <§> Displa
I O Displa
1 O PlotAi

ly extreme values and their times

iy Spring Forces at Time: 0.0000

Loop Pile Grou|

limated M Ml M W POOOO

p Node Index [l

▼ | Degree of Freedom: (x ▼]

Long.

Trans.

Node

Force

Time

Force

Time

Moment

Time

(kips)

(sec)

(kips)

(sec)

(kip-ft)

(sec)

251.4489

0.21

0.9333

0.142

0.0012

0.0047

0.21

0.001

0.18

0.0138

0.21

0.0028

0.18

0.0229

0.21

0.0045

0.18

0.032

0.21

0.006

0.18

0.0411

0.21

0.0073

0.178

0.0502

0.21

0.0083

0.178

0.0593

0.21

0.009

0.178

0.0684

0.21

0.0093

0.178

0.0775

0.21

0.0093

0.178

0.0866

0.21

0.0089

0.176

0.0957

0.21

0.0082

0.176

0.1048

0.21

0.0078

0.086

0.1139

0.21

0.0082

0.082

0.123

0.21

0.0088

0.08

0.1321

0.21

0.0096

0.078

0.1412

0.21

0.0105

0.076

0.1505

0.21

0.0092

0.076

0.1596

0.21

0.0059

0.072

0.1687

0.21

0.0059

0.14

0.1778

0.21

0.007

0.134

0.1869

0.21

0.0087

0.13

0.196

0.21

0.0105

0.126

0.2051

0.21

0.0121

0.124

0.2143

0.21

0.0134

0.124

0.2234

0.21

0.0144

0.122

0.2325

0.21

0.0148

0.12

0.2416

0.21

0.0149

0.118

ERDC/ITL TR-16-1

140

Figure 5.36 Pile Group Response Peak Deflections.

Figure 5.37 Pile Group Response Maximum and Minimum Forces and Moments.

ERDC/ITL TR-16-1

141

When the “Plot Animated” radio button is selected, the user can plot an
individual pile group node’s force-versus-displacement over time. Using
the “Pile Group Node Index” combo box, the user can select a pile group
node for display. The “degree of freedom” combo box provides the axis
that the resulting force is applied. The animation control behaves in the
same manner as it did for the FEO Nodal Output window.

When these options have been selected, the graphic window below shows
the relevant plot (Figures 5.38 through 5.40). At the top of the display is
the current time, the selected pile group node, and selected force axis. To
the left of the plot is a display of the coordinate system and the wall with
nodes and element connectivity. The selected pile group node is
highlighted in red to show the pile group node position relative to the
structure. The plot of the impact-load time history from the starting
position of the impact to the ending impact location, based on the velocity
of the impacting barge train, is displayed along the wall. The current
location of the load is displayed as a red line along the time history.

Figure 5.38 Animated plot of node 85 X-force and displacement vs time.

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142

Figure 5.39 Animated plot of node 85 Y-force and displacement vs time.

Pile Group Response

lmpact_Deck_TestProbleml .feo

O Display extreme values and their times

Plot Animated MHTHIUMliHl 0.26000 _

|4J| g] Loop Pile Group Node Index: 185 ▼] Degree of Freedom: hr

Time = 0.26000 Node: 85 Y-Axis

Figure 5.40 Animated plot of node 85 Z-force and displacement vs time.

Pile Group Response

File

lmpact_Deck_TestProbleml feo

1 O Display extreme values and their times

1 O Display Spring Forces at Time: 0.2600

1 ® Plo, Animated | IjlNll^l Mil MlW] 0.26000

g] Loop Pile Group Node Index: [85 ▼ ] Degree of Freedom: [z ▼]

Time = 0.26000 Node: 85 Z-Rotation

0.25

0.2

Force (kips)

0.05

0.04

0.03

0.02

0.01

0 . 0 -

- 0.01

- 0.02

-0.03

-0.04

-o.g

Force (kips)

- 0.1

0.005 -0.0025 0.0 0.0025 0.005

Displ. (in)

- 0.2

- °- 2 5.0 1.0 2.0 3.0 4.0 5.0

Time (sec)

0.25

0.2 1

0.1

0 . 0 -

- 0.1

- 0.2

~°- 2 §.0 1.0 2.0 3.0 4.0 5.0

Time (sec)

ERDC/ITL TR-16-1

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Besides the wall plot, there are three additional plots. The plot in the
upper left is the resultant force-versus-time plot. This force results from
spring deformation along the selected axis. A red point shows the current
time and its force for the selected pile group node.

The plot in the lower left is the displacement versus time plot. A red point
shows the current time and its displacement for the selected pile group
node.

The center plot shows the force-versus-displacement plot. This plot shows
how force varies with displacement at the pile group node. Because this
graph has axis with values that may be repeated (and therefore is not a
mathematical function), the current force and time are located with cross¬
hairs for the current time.

Note that the Z-axis DOF is not a 3-D axis but a rotational axis at the pile.
For pile group nodes that are not connected due to a moment release
between deck sections, the force versus the rotational displacement will
give a zero value.

5.7.4 Run Information

Pressing the Show Run Output button from the Output Tab brings up a
text window with the results of the program run (Figure 5.41). The menu
allows the user to save the file to a new location or exit the Run Output
window. How the program worked is displayed in this window, as well as
any specific finite element data requested in the Analyze Tab.

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Figure 5.41 Output run information with selected FEO node output.

[mpact_Deck_TestProbleml.out - — 4

File

Maximum applied

normal force

516.442073

Maximum applied

tangent force =

258.221037

Time for maximum applied forces =

0.200000

Located at x =

501.389500

Spring "Y"

Maximum

Time max.

Maximum

Time max.

at Node Displacement

Displ.

Force

0.0000

0.2100

0.0047

0.2100

0.0000

0.2100

0.0138

0.2100

0.0001

0.2100

0.0229

0.2100

0.0001

0.2100

0.0320

0.2100

0.0001

0.2100

0.0411

0.2100

0.0001

0.2100

0.0502

0.2100

0.0002

0.2100

0.0593

0.2100

0.0002

0.2100

0.0684

0.2100

0.0002

0.2100

0.0775

0.2100

0.0002

0.2100

0.0866

0.2100

0.0003

0.2100

0.0957

0.2100

0.0003

0.2100

0.1048

0.2100

0.0003

0.2100

0.1139

0.2100

0.0003

0.2100

0.1230

0.2100

0.0004

0.2100

0.1321

0.2100

0.0004

0.2100

0.1412

0.2100

0.0004

0.2100

0.1505

0.2100

0.0005

0.2100

0.1596

0.2100

0.0005

0.2100

0.1687

0.2100

0.0005

0.2100

0.1778

0.2100

0.0005

0.2100

0.1869

0.2100

0.0006

0.2100

0.1960

0.2100

0.0006

0.2100

0.2051

0.2100

0.0006

0.2100

0.2143

0.2100

0.0006

0.2100

0.2234

0.2100

0.0007

0.2100

0.2325

0.2100

0.0007

0.2100

0.2416

0.2100

0.0007

0.2100

0.2507

0.2100

0.0007

0.2100

0.2598

0.2100

0.0008

0.2100

0.2689

0.2100

□

5.8 Example: Geometry Input for the Impact Deck at Lock and
Dam 3

The example problem for the Lock and Dam 3 impact deck that was input
in section 2 present the most complex geometry for GUI input. In this
case, the approach wall structure had 8 reinforced concrete monoliths.
These monoliths were connected to each other and with the single end cell
at the start of the approach wall with a pinned connection, and to the
neighboring monoliths through a connection that transfer shear and axial
forces but not moments.

The monoliths themselves were to be constructed by connecting together
8 concrete segments that were 12 ft 6 in. in length. Each segment was
supported by two pile groups. Each pile group cluster consisted of 3 piles -
a vertical pile at the front of the group, closest to where an impact would
occur, and two batter piles with a batter of 1:4. The spacing between the pile
groups was to be 6 ft 3 in., which resulted in monoliths with a length of
100 ft. These plans changed, as engineering plans often do, due to wall
length requirements. The new monolith length ended up becoming 104 ft
10 in. The distance between pile groups grew to 6 ft 3.5 in. The distance
from the ends of the monolith to the first and last pile group was 3 ft 4.25 in.

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In this case, it was natural to model each monolith as a set of nodes in the
Impact_Deck GUI, and then copy the monolith nodes 7 times (for 8
monoliths). The end nodes of the first monolith were the connecting nodes.
These nodes were at o ft o in. and 104 ft 10 in., which translated to decimal
feet of 0.0 ft and 104.833 ft. Because inches did not convert to nice decimal
feet, the user must determine his or her personal level of precision. Because
that last node was coincident with the first node of the next monolith, the
last node does not needed to be entered as it will be created in the copy and
offset command. Therefore, the user only needs to place a node at position
0.0 ft. Because it was an inter-monolith node, the check-box was checked to
flag the node (Figure 5.42).

The rest of the nodes in the monolith represented each pile group location
in the monolith. These monolith groups started 3 ft 4.25 in. from the start
of the monolith (at o ft o in.) and ended at 3 ft 4.25 in. from the end of the
monolith (at 104 ft 10 in.). These coordinates in decimal feet were 3.354 ft
and 101.479 ft, respectively. There were 16 pile groups in the monolith
with 15 divisions between them. Entering these values into the
“Interpolated Node Input” box (with the inter-monolith nodes check-box
unchecked) created the nodes for the pile groups (Figure 5.43).

Figure 5.42 Add node at position 0.0 ft as an inter-monolith node.

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Figure 5.43 Add interpolated nodes from 3.3541666 ft to 101.4791666 ft.

When the set of nodes for the monolith were created (without the ending
inter-monolith nodes), all of the nodes for the monolith were selected using
a left-mouse, click-drag operation (Figures 5.44 and 5.45). Clicking the
“Copy Selected Nodes...” button allowed the user to create copies of this
monolith node set multiple times (Figure 5.46). Enter in the offset of the
monolith as 104 ft 10 in. (104.833 decimal feet- Figure 5.47), then click the
OK button in the “Copy Selected Nodes with Offset” window seven times
(Figure 5.48).

Figure 5.44 Selecting nodes with a left-mouse, click-drag.

Figure 5.45 Selected nodes are shown with vertical lines.

Input Plot

r Y

o Pile Group Node

Node 16
YPos:

94.9375000003

333

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Figure 5.46 Selected nodes can be copied multiple times with the copy selected

nodes button.

Figure 5.47 The copy selected nodes dialog lets the user specify

an offset.

Figure 5.48 Select OK the number of times that the
selected nodes need to be copied.

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Because the monolith was modeled without the ending inter-monolith
connection, the last connection was added using the “Single Node Input”
box. Its position was at 838’-8” (838.666 decimal feet). This node was
created as an inter-monolith node (Figure 5.49).

Figure 5.49 Finally, Add the Final Node.

Hopefully, this subsection has revealed the usefulness of the Impact_Deck
GUI methods for modeling geometry using interpolation and copying of
repetitive structures that might otherwise require a good amount of
calculation for the user.

5.9 Final Remarks

In this section, the user was presented with the GUI specifications. From
this information, the user should be able to define the geometry, select a
force time history for an impact, give beam and pile group properties, and
perform an analysis from this input model. After the analysis was
performed, methods for varying visualization of results were presented,
either as static or dynamic plots with information for nodes, elements, and
resultant forces at pile groups. The user was also presented with examples
that show input for the various structures, and the GUI output (impact
decks, flexible approach walls, and guard walls).

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6 Conclusions and Recommendations

In this research project, an involved dynamic time-domain analysis was
performed to determine the displacement and response forces of flexible
pile groups supporting a concrete beam or deck used for the absorption of
energy from a barge train impact. An impact force time history was used to
represent an impact event (Ebeling et al. 2010). Three different types of
pile-founded, flexible approach walls were studied: an impact deck, an
alternative flexible approach wall, and a guard wall.

The first case was the Lock and Dam 3 impact deck structure consisting of
eight concrete monoliths. These monoliths were supported over a series of
equally spaced rows of three cluster pile groups. An internal pin (i.e., with
no bending moment transfer) formed the connection between adjacent
monoliths. At one end of the structural system, the impact deck monolith
was pin-connected to a massive concrete circular cell and at the other end
of the structural system the impact deck monolith was free. This structural
system was modeled by means of typical beam elements between each
pile-group row or between the last pile group row of the monolith to the
inter-monolith pin connection. Each pile-group was modeled by using a
pair of elastic-plastic translational springs. The definitions of the spring
stiffness was determined by doing a push-over analysis of a single pile-
group cluster. The dynamic-impact load had a specified starting point and
was stationary or moving along the wall. A damping effect was included
using the Rayleigh damping theory, which depends on the natural
frequency of the system. The two natural frequencies of the structural
system (to be used for Rayleigh damping) were calculated in an
approximate form by using the spring stiffness and the mass of the impact
beam or deck. After the global mass, damping, and stiffness matrices and
the load vector were assembled, the dynamic time-history response of the
system was calculated.

The second case was the McAlpine alternative flexible approach wall. A
section of the approach wall consisted of two consecutive concrete beams
supported over three pile groups was modeled with the software. The
effect of non-impacted beams was modeled simplistically and the user
specifies how many beams before and after the area of interest exist (i.e.,
where the impact event occurs). The connection of the first beam to the
second beam at the center pile cap was accomplished by using a shear key
(i.e., no bending moment transfer). In fact, due to the distance between

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the end of one beam and the start of the second beam, the shear key had to
be modeled also as two rigid beam elements, which connected at a node
centered between the two beam end nodes. From there, a rigid beam
element was perpendicularly connected back to a node at the center of
rigidity of the pile group and its cap. This established the position of the
two translational springs and one rotational spring support. In this way,
the force-translational and moment-rotational effects of the central pile
group were modeled. This structural system was modeled by means of
typical beam elements between each pile group and the three rigid beam
elements of the central pile cap. The pile groups at the start and end of the
structural system were modeled by using elastic-plastic translational
springs. The central pile group was modeled by using two elastic-plastic
translational springs and one elastic-plastic rotational spring. The
definitions of the translational spring stiffness was determined by doing a
push-over analysis of a typical pile group. The rotational spring properties
were defined by using the translational spring properties of the pile group
as shown in Appendix E. The dynamic impact-force time history had a
specified starting point and was stationary or moving along the beam. A
damping effect was included using the Rayleigh damping theory, which
depended on the natural frequency of the system. The two natural
frequencies of the structural system (to be used for Rayleigh damping)
were calculated in an approximate form by using the spring stiffness and
the mass of the beam. After the global mass, damping, and stiffness
matrices and the load vector were assembled, the dynamic time-history
response of the system was calculated.

The third case was a typical guard wall. A section of the approach wall
consisted of two consecutive concrete beams supported over three pile
groups was modeled with the software. The connection of the first beam to
the second beam in the center pile cap was achieved by using a shear key
(i.e., no bending moment transfer). This structural system was modeled by
means of typical beam elements between each pile row. The beam
elements that connect at the central pile row were modeled with end
releases (i.e., no moment transfer). The start, central, and end pile rows of
the structural system were modeled by using a pair of elastic-plastic
springs. The definitions of the stiffness for the translational and
longitudinal springs were determined by doing a push-over analysis of a
typical pile group. The dynamic impact load had a specified starting point
and was stationary or moving along the beam. A damping effect was
included using the Rayleigh damping theory, which depended on the

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natural frequency of the system. The two natural frequencies of the
structural system (to be used for Rayleigh damping) were calculated in an
approximate form by using the spring stiffness and the mass of the impact
beam or deck. After the global mass, damping, and stiffness matrices and
the load vector were assembled, the dynamic response of the system was
calculated.

The general input data for these three models were the nodes and their
positions along the wall, modulus of elasticity of the beam, cross-sectional
area of the beam, moment of inertia of the beam, mass per unit length of
the beam, damping ratio, dynamic coefficient of friction between the lead
impact barge and the wall, the initial point of contact, velocity of the
moving load, springs properties, and force time history description. The
accuracy of the solution depended on the appropriateness of these
variables.

To demonstrate the effectiveness of the methodology developed in the
Impact_Deck computer program, a validation against SAP2000 and
several examples were presented. The validation produced outstanding
results. In the examples when the plastic behavior was reached, the time
history results were in agreement with the developed elastic-plastic, force-
displacement relationship of the springs. It was recognized that SAP2000
will not handle a moving dynamic load nor the nonlinear springs used to
model the unique clustered pile groups responses. These are two unique
capabilities of the Impact_Deck software.

An important conclusion from the Impact_Deck analyses was that inertial
effects of the wall superstructure and substructure during dynamic loading
were important to the computed results and should not be ignored. A
dynamic analysis must be performed because the resulting overall peak
response force was much greater than the peak input impact force and the
overall peak response force happened much later in time than when the
peak input impact force occured. For every structure analyzed, the overall
peak response force had a greater value than the peak input impact force
value and the time to the overall peak response force was greater than the
time to the peak input impact force. For the simply supported beam guard
wall model (in section 4) the overall response force was 181% of the peak
input impact force. The overall peak response time was 0.392 sec, or
0.192 sec after the peak input impact force time of 0.2 sec.

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The peak response force for any single pile-group node as a substructure
for a simply supported beam, in situations where load sharing was
minimal (in the McAlpine alternative flexible approach wall and guard
wall examples), also exceeded the peak input load force and, due to inertial
effects, happened at a later time than the peak input impact force. For the
McAlpine alternative flexible approach wall and the guard wall examples,
the peak response force for the single pile group node with the greatest
force was and 123% and 128% of the peak input load force, respectively.
These peaks occurred 0.08 and 0.132 sec after the time to peak input
impact force. These were important considerations for the design of pile
supported approach wall structures subjected to barge train impact
loading.

The peak response force for any single pile-group node with a fixed
connection to a deck, and where multiple pile groups support the monolith
deck, benefited from the effect of load sharing. In this case, the motion of
the impact deck generated a response from all of the pile groups (16, in our
example). Because of the shear key connection between monoliths, the pile
groups of all 8 monoliths responded similarly to the impact deck motion.
However, despite load sharing at 0.04 sec after the peak input load, the
peak response force for any pile group reached its peak value of 56.456
kips. This was 11% of the peak input load force, or 7.8% of the peak overall
pile-group response force. While it was reasonable to assume a force
reduction of greater than 16 times could occur, the real reduction was only
slightly more than 9 times less for the peak input load and nearly 13 times
less for the peak overall pile group response. Load sharing occured, but its
effects were not as pronounced, due to the inertia of the impact deck under
a dynamic barge train time history load. These results demonstrated the
advantage of using this moment resistant impact deck monolith supported
on a large number of smaller piles versus using a simply supported impact
beam with larger piles and longer spans: the design forces for each
individual pile group was much less. Site conditions and the use of clever,
cost-effective, in-the-wet and above-the-wet construction practices
(including the use of precast structural features) will dictate which design
type will ultimately possess the greater advantage (in cost and effort) for
an approach wall project.

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References

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Chopra, Anil K. 1995. Dynamics of structures; theory and applications to earthquake
engineering, Englewood Cliffs, New Jersey: Prentice Hall.

Chopra, Anil K. 2001. Dynamics of structures; theory and applications to earthquake
engineering, second edition, Englewood Cliffs, New Jersey: Prentice Hall.

Clough, Ray W., and Joseph Penzien. 1993. Dynamics of structures, Second edition, New
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Computers and Structures, Inc. 2003. SAP2000 static and dynamic finite element
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Craig, R. R. 1981. Structural dynamics, an introduction to computer methods, New York,
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Davisson, M. T. 1970. Lateral load capacity of piles, Highway Research Record, Number
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Ebeling, Robert M., R. A. Green, and S. E. French. 199 y. Accuracy of response of single

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Ebeling, Robert M., Barry C. White, Abdul N. Mohamed, and Bruce C. Barker. 2010.

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lock wall using impact^force, ERDC/ITL TR-10-1, Vicksburg, MS: U.S. Army
Engineer Research and Development Center.

Ebeling, Robert M., Abdul N. Mohamed, Jose R. Arroyo, Barry C. White, Ralph W. Strom,
and Bruce C. Barker. 2011. Dynamic structural flexible-beam response to a
moving barge train impact force time-history using impact_beam, ERDC/ITL
TR-11-1, Vicksburg, MS: U.S. Army Engineer Research and Development Center.

Ebeling, R. M., R. W. Strom, B. C. White, and K. Abraham. 2012. Simplified Analysis

Procedures for Flexible Approach Wall Systems Founded on Groups of Piles and
Subjected to Barge Train Impact, ERDC/ITL TR-12-3, Vicksburg, MS: U.S.
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Hartman, J. P., Jaeger, J. J., Jobst, J. J., and Martin, D. K. 1989. User's Guide: Pile

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McGuire, W., and R. H. Gallagher. 1979. Matrix structural analysis, New York, NY: John
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Patev, Robert C., Bruce C. Barker, and Leo V. Koestler. 2003. Prototype barge impact
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Press, W. H., S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery. 1996. Numerical

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Reddy, J. N. 1993. An introduction to the finite element method, Second edition,
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Ross, C. T. F. 1991. Finite element programs for structural vibrations, New York, NY:
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Stadler, W., and R. W. Shreeves. 1970. The transient and steady-state response of the

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Appendix A: Lock and Dam 3 - Equations of
Motion for the Mathematical Model

In structural dynamics, the mathematical model of bodies of finite
dimensions undergoing translational motion are governed by Newton’s
Second Law of Motion, expressed as

^F = m«a (1.1 bis)

at each time step t during motion. In the mathematical model of
transverse vibration, the forces acting on the flexible impact beam mass at
each time step t include (l) the impact force at time step t, (2) the elastic
restoring forces (of the beam), and (3) the damping forces (of the beam).
The mathematical model of the beam in the engineering formulation
described in this appendix has a finite number of degrees of freedom
(DOF) because it is discretized using the finite element formulation. The
engineering formulations of equations of motion are solved using a
numerical solution method to determine the displacement and response
forces at each pile bent support feature. Each group of clustered piles is
modeled as transverse and longitudinal elastic/plastic springs. This is a
dynamic process, with the response forces responding to an impact pulse
force time history and each calculation of the equations of motion
occurring at each time step. The applied impact force can be given a
constant velocity parallel to the approach wall so that it is changing
position with time. The responses of the elastic beam over elastic or plastic
springs can be obtained by the use of the Multi-Degrees-of-Freedom
model (MDOF) and the finite element formulation of the beam element.
The Impact_Deck software includes damping forces by using the Rayleigh
damping model. The response is obtained using either the HHT-a method
(Appendix F) or the well-known Wilson -0 method (Appendix G). By using
the Newton’s Second law and applying it to a MDOF system, the resulting
equations of motion can be expressed as,

[Af]{ii(t)} + [C]{li (t)} + [*]{u(t)} = (F(t)} (2.1 bis)

where,

[M] = global mass matrix
[C] = global damping matrix

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[K] = global stiffness matrix
(F(t)} = global vector of external forces and moments
(u(t)}, (u(t)}, {u(t)} = relative acceleration, relative velocity, and relative
displacement for each DOF.

This appendix discusses the various relationships used in this engineering
methodology that are implemented in Impact_Deck.

A.l Element degrees of freedom (DOF) and interpolation functions

Consider a straight-beam element of length L, mass per unit length m(x),
and flexural rigidity El (x). The two nodes by which the 2-D finite element
can be assembled into a structure are located at its ends. If only planar
displacements are considered, each node has three DOF: the longitudinal
displacement, the transverse displacement, and rotation.

The longitudinal displacement (i.e., axial direction) of the beam element is
related to its two DOF:

U ( X > t ) = Yj U i(. t W X ) (A- 1 )

1=1

where the function <fii(x) defines the displacement of the element due to
unit displacement u it while constraining other DOF to zero. Thus (pi(x)
satisfies the following boundary conditions and are shown in Figure A.i,

/ = 1:%(0) = 1, tp ± (L) = 0

(A. 2 )

/ = 2 :\p 2 ( 0 ) = 0 , W 2 (L)= 1

(A. 3 )

Note that these DOF subscript values i = l and 2 correspond to Figure A.i
DOF subscripts i = 1 and 4.

The transverse displacement and rotation of the beam element is related
to its four DOF,

u(x,t) = Yjti(t)Wi(x) i = 2,3,5,6

(A.4)

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where the function ify(x) defines the displacement of the element due to
unit displacement u i: while constraining other DOF to zero. Thus, ify(x)
satisfies the following boundary conditions

Figure A.1 Shape function for axial displacement
effect.

i = 1: iM0)=l, (0) =% (L) =Wi (L) = 0 (A.5)

i = 2 : ip'fOj = 1, ip 2 (0) =ip 2 (L) = \p 2 (L) = 0 (A.6)

/ = 3 : ip 3 (. L ) = l,ip 3 (0) = ip 3 (0) = ip 3 (L) = 0 (A.7)

i = 4:ip\(L) = l,ip 4 (0)=ip' 4 (0)=ip 4 (L) = 0 (A.8)

Note that these DOF subscript values i = l, 2, 3 and 4 correspond to
Figure A.2 DOF subscripts i = 2, 3, 5, and 6.

These interpolation functions could be any arbitrary shapes satisfying the
boundary conditions. One possibility is the exact deflected shapes of the
beam element due to the imposed boundary conditions, but these are
difficult to determine if the flexural rigidity varies over the length of the
element. However, they can conveniently be obtained for a uniform beam as
illustrated next for the transverse displacement and rotation. Neglecting
shear deformations, the equilibrium equation for a beam loaded only at its
ends is

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158

„ T d A u .

£/ d7 = ° (A ‘ 9)

The general solution of Equation (A.9) for a uniform beam is a cubic
polynomial

+ flo

/ \

+ a a

'x'

(A.10)

The constants a t can be determined for each of the four sets of boundary
conditions of Equations (A.5) to (A.8), to obtain

Ti(*) = l - 3

\y 2 (x) = L

( ^

1 X

+ 2

( \3

1 X '

( \

' x'

-2 L

+ L

~L,

/ \

— z

T 3 W = 3

W 4 (x) = -L(x/Lf+L(x/Lf

(A.ll)

(A.12)

(A.13)

(A.14)

These interpolation functions, illustrated in Figure A.2, can be used in
formulating the element matrices. The same process can be done with the
axial effect to obtain the basic interpolation functions,

(A.15)

M> 2 ( x ) = §

(A.16)

The finite element method is based on assumed relationships between the
displacements at interior points of the element and the displacements at
the nodes. Proceeding in this manner makes the problem tractable but
introduces approximations in the solution.

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Figure A.2 Shape function for transverse
displacement and rotation effect.

A.2 Element stiffness matrix

Consider a beam element of length L with flexural rigidity EI(x). By
definition, the stiffness influence coefficient /q ; - of the beam element is the

force in the DOF i due to unit displacement in DOFj. Using the principle
of virtual displacement, the general equation for k t j, which is the stiffness

term for the transverse displacement and rotation, in the element stiffness
matrix is

(A. 17 )

The symmetric form of this equation shows that the element stiffness
matrix is symmetric, k t j = kj t . Equation (A.17) is a general result in the

sense that it is applicable to elements with arbitrary variation of flexural
rigidity EI(x), although the interpolation functions of Equations (A.11) to
(A.14) are exact only for uniform elements. The associated errors can be
reduced to any desired degree by reducing the element size and increasing
the number of finite elements in the structural idealization. For a uniform
finite element with EI(x) = El, the integral of Equation (A.17) can be
evaluated analytically for i,j = 2,3,5, and 6, resulting in the
corresponding terms in the element stiffness matrix.

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In the same way, for the axial contribution, the general equation for /qy,
which is the stiffness term for the axial displacement in the element
stiffness matrix is

ky = fEA(x)\p](x)\p'j(x)dx

(A.18)

For a uniform finite element with EA(x) = EA, the integral of Equation
(A.18) can be evaluated analytically for i,j = 1,4 resulting in the
corresponding terms in the element stiffness matrix. Finally, when all
terms are calculated, the element stiffness matrix can be obtained as,

K\ =

Y 1 EI

6EI

12 EI

6EI

L 3

L 2

L 3

L 2

6EI

L 2

4 El

6EI

L 2

2 EI

12 EI

6EI

12 EI

6 El

L 3

L 2

L 3

L 2

6EI

2 EI

6EI

4 EI

L 2

(A.19)

These stiffness coefficients are the exact values for a uniform beam,
neglecting shear deformation, because the interpolation functions are the
true deflection shapes for this case. Observe that the stiffness matrix of
Equation (A. 19) is equivalent to the force-displacement relations for a
uniform beam that are familiar from classical structural analysis.

A.3 Member end-releases

When a structure has an internal pin (i.e., no moment transfer from one
element to the adjacent element), the DOF associated to the rotation must
have a stiffness value of zero for that DOF. In that way, the element will
keep the ability to transfer the axial and shear force but not the bending
moment. That process of assigning a zero value to one term in the stiffness
matrix will affect the other terms because equilibrium has to be
maintained.

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For a beam element, the six equilibrium equations in the local reference
system can be written as

fij= k ij u ij (A.20)

If one end of the member has a hinge, or other type of release that causes
the corresponding force to be equal to zero, Equation (A.20) requires
modification. A typical equation is of the following form:

fn=Yh u i

j =1

(A.21)

If we know a specific value of f n is zero because of a release, the
corresponding displacement u n can be written as

n—1 h- 12 b

u = V^U.+ V u +r

n Z_^ 7 j 1 7 1 'n

j= 1 ^ nn j=n+ 1 ^ nn

(A.22)

Therefore, by substitution of Equation (A.22) into the other five
equilibrium equations, the unknown u n can be eliminated and the
corresponding row and column set to zero, or

fij = ki j u ij +n j

The terms f n = 0 and the new stiffness terms are equal to

kij

(A.23)

(A.24)

This procedure can be repeatedly applied to the element equilibrium
equations for all releases. The repeated application of the simple
numerical equation is sometimes called the static condensation or
partial Gauss elimination.

There is a special case when the load is applied at the end release node. In
this case, the load must be altered to maintain the o moment transfer. This
special case is discussed in Appendix H.

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A.4 Element mass matrix

The mass influence coefficient for a structure is the force in the i th DOF
due to unit acceleration in the j th DOF. Applying this definition to a beam
element with distributed mass m (x) and using the principle of virtual
displacement, a general equation for can be derived:

m ij= J m { x )Wi{x)Wj{x)dx (A.25)

The symmetric form of this equation shows that the mass matrix is
symmetric; . If we use the same interpolation functions of

Equations (A. 11 ) to (A. 16 ) as were used to derive the element stiffness
matrix into Equation (A. 20 ), the result obtained is known as the consistent
mass matrix. The integrals of Equation (A. 20 ) are evaluated numerically
or analytically depending on the function m(x). For an element with
uniform mass per unit length (i.e., m(x) = m), the integrals can be
evaluated analytically to obtain the element (consistent) mass matrix as

m e = mL

156

22 L

— 13L

420

22 L

4L 2

13L

420

13L

156

-22 L

420

— 13L

-22 L

4L 2

420

420 .

(A.26)

A.5 Element (applied) force vector

If the external forces p*(t), i = 1,2, 3,4,5, and 6 are applied along the six
DOF at the two nodes of the finite element, the element force vector can be
written directly. On the other hand, if the external forces are concentrated
forces p 'j(t) at locations Xj, the nodal force in the i th DOF is

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p,M = X»,<V < a - 27 >

This equation can be obtained by the principle of virtual displacement. If
the same interpolation functions of Equations (A. 11 ) to (A. 16 ), are used to
derive the element stiffness matrix as used here, the results obtained are
called consistent nodal forces.

A.6 Nonlinear force-deflection relationship for the springs supports

The software Impact_Deck has the capability to calculate the response of
spring supports if the springs develop plastic behavior in the force-
displacement relationship. The spring can be considered as linear if the
load in the spring is below the elastic displacement Seias and the elastic
force Feias as shown in Figure A. 3 . If the load is reduced and the force-
displacement is below point 1 , the unload returns along the same path as
the loading phase. The loading phase is shown using green arrows and the
unloading phase is shown using red arrows. However, if the load is greater
than the elastic displacement and it is in the loading stage, it follows the
green arrows until reaching the maximum force-displacement, point 2 . If
the unload occurs from this point, it will unload following a slope specified
by the user. In this case, the slope proceeds from point 2 to point 4 . If the
force never increases to point 2 again, the force-displacement will remain
along the line from point 2 to point 4 . If the force decreases to point 4 , zero
force is reached with a plastic permanent deflection. If the load increases
again until point 2 is reached, the original backbone is rejoined,
proceeding from point 2 towards point 3 . If the force reaches a maximum
on the line between point 2 and 3 and starts to decrease again, the load-
deflection will follow the same unload slope as the slope from point 2 to
point 4 but starting from the new maximum force-deflection. If the force-
deflection is greater than point 3 , Impact_Deck assigns a zero value to this
spring because the maximum value was reached and failure occurs.

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Figure A.3 Force-displacement relation of the spring
support.

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Appendix B: McAlpine Alternative Flexible
Wall - Equations of Motion for the
Mathematical Model

In structural dynamics the mathematical model of bodies of finite
dimensions undergoing translatory motion are governed by Newton’s
Second Law of Motion, expressed as

^F = m«a (1.1 bis)

at each time step t during motion. In the mathematical model of transverse
vibration, the forces acting on the flexible impact beam mass at each time
step t include (l) the impact force at time step t, ( 2 ) the elastic restoring
forces (of the beam), and ( 3 ) the damping forces (of the beam). The
mathematical model of the beam in the engineering formulation described
in this section has a finite number of degrees of freedom (DOF) because it is
discretized using the finite element formulation. The engineering
formulation of equations of motion are solved using a numerical solution
method to determine the displacement and response forces at each pile bent
support feature. Each group of clustered piles is modeled as transverse,
longitudinal, and rotational elastic/plastic springs. This is a dynamic
process, with the response forces responding to an impact pulse force time
history and each calculation of the equations of motion occurring at each
time step. The applied impact force can be given a constant velocity parallel
to the approach wall so that it is changing its position with time. The
responses of the elastic beam over elastic or plastic springs can be obtained
by the use of the multiple degrees of freedom model (MDOF) and the finite
element formulation of the beam element. Impact_Deck includes the
damping forces by using the Rayleigh damping model. The response is
obtained using the HHT- a method (Appendix F) or the well-known
Wilson-0 method. By using the Newton’s Second law and applying it to a
MDOF system, the resulting equations of motion can be expressed as

[M]{u(t)J + [C]{u (f)} + [q{u(t)} = {F(0} (2.1 bis)

where:

[M] = global mass matrix

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[C] = global damping matrix
[K] = global stiffness matrix
{F(t)} = global vector of external forces and moments
(u(t)}, {it(t)} = relative acceleration, relative velocity, and relative
displacement for each DOF.

This appendix discusses the various relationships used in this engineering
methodology that are implemented in Impact_Deck.

B.l Element degrees of freedom (DOF) and interpolation functions

Consider a straight-beam element of length L, mass per unit length m(x),
and flexural rigidity El (x). The two nodes by which the 2 -D finite element
can be assembled into a structure are located at its ends. If only planar
displacements are considered, each node has three DOF: the longitudinal
displacement, the transverse displacement, and rotation.

The longitudinal displacement (i.e., axial direction) of the beam element is
related to its two DOF:

(B- 1 )

1=1

where the function defines the displacement of the element due to

unit displacement u t while constraining other DOF to zero. Thus, 0;(x)
satisfies the following boundary conditions and are shown in Figure B.l,

i=l:\V 1 (0) = l,\\) 1 (L) = 0 (B.2)

/ = 2 : \p 2 (0) = 0, ip 2 (L) —1 (B.3)

Note that these DOF subscript values i = l and 2 correspond to Figure B.l
DOF subscripts i = l and 4 .

The transverse displacement and rotation of the beam element is related
to its four DOF,

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Figure B.l Shape function for axial displacement
effect.

u(x,t) = jyi(t)Wi(x) i = 2 ,3,5,6

(B.4)

where the function i/>;(x) defines the displacement of the element due to
unit displacement Ui, while constraining other DOF to zero. Thus, ify(x)
satisfies the following boundary conditions,

i= 1 •' tpfyOJ = l,ip 1 (0) =tp ± (L) =\Pi (L) = 0

(B.5)

i= 2 :tp' 2 ro; = i,\p 2 ( 0 )=ip 2 (L) = \p 2 (L) = 0

(B. 6 )

i= 3 : tp 3 (L) = (0) = W 3 ( 0 ) = W 3 {L) = 0

(B.7)

i= 4 ■’W , 4( l ) = 1 ’W4( 0 )=W , 4(0)=W 4 ( l ) = 0

(B. 8 )

Note that these DOF subscript values i = l, 2 , 3 , and 4 correspond to
Figure B .2 DOF subscripts i = 2 , 3 , 5 , and 6 .

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as illustrated next for the transverse displacement and rotation. Neglecting
shear deformations, the equilibrium equation for a beam loaded only at its
ends is

d 4 u

dx 4

(B.9)

The general solution of Equation (B. 9 ) for a uniform beam is a cubic
polynomial

u(x) =a 1 +a 2

+ flo

/ \

+ a a

(B.10)

The constants a t can be determined for each of the four sets of boundary
conditions of Equations (B. 5 ) to (B. 8 ), to obtain

xpi(x) = l - 3

W 2 ( x ) = l
W 3 (*) = 3
T 4 {x) = -L

( \

-2 L

+ L

( \

_ 9

/ \

— z

yL,

+ L

'x ' 3

(B.ll)

(B.12)

(B.13)

(B.14)

These interpolation functions, illustrated in Figure B. 2 , can be used in
formulating the element matrices. The same process can be done with the
axial effect to obtain the basic interpolation functions,

t 2 (*) = -

(B.15)

(B.16)

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Figure B.2 Shape function for transverse
displacement and rotation effect.

B.2 Element stiffness matrix

Consider a beam element of length L with flexural rigidity EI(x). By
definition, the stiffness influence coefficient k t j of the beam element is the

force in the DOF i due to unit displacement in DOFj. Using the principle
of virtual displacement, the general equation for k^, which is the stiffness

term for the transverse displacement and rotation, in the element stiffness
matrix is:

k, = /EI( x)ip ](x)Wj(x)dx (B.17)

The symmetric form of this equation shows that the element stiffness
matrix is symmetric; /q ; - = kj t . Equation (B. 17 ) is a general result in the

sense that it is applicable to elements with arbitrary variation of flexural
rigidity EI(x), although the interpolation functions of Equations (B. 11 ) to
(B. 14 ) are exact only for uniform elements. The associated errors can be
reduced to any desired degree by reducing the element size and increasing
the number of finite elements in the structural idealization. For a uniform
finite element with EI(x) = El, the integral of Equation (B. 17 ) can be

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B.3

evaluated analytically for i,j = 2,3,5, and 6, resulting in the
corresponding terms in the element stiffness matrix.

In the same way, for the axial contribution, the general equation for k t j,
which is the stiffness term for the axial displacement in the element
stiffness matrix is

ky = fEA(x)\p](x)\p'j(x)dx (B. 18 )

For a uniform finite element with EA(x) = EA, the integral of Equation
(B.18) can be evaluated analytically for i,j = 1,4 resulting in the
corresponding terms in the element stiffness matrix. Finally, when all
terms are calculated, the element stiffness matrix can be obtained as

12 EI

6EI

12 EI

6 El

L 3

L 2

L 3

L 2

*1=

6 El

L 2

4 El

6EI

L 2

2 EI

12 EI

6EI

12 EI

6 El

L 3

L 2

L 3

L 2

6EI

2 EI

6EI

4 EI

L 2

(B. 19 )

These stiffness coefficients are the exact values for a uniform beam,
neglecting shear deformation, because the interpolation functions are the
true deflection shapes for this case. Observe that the stiffness matrix of
Equation (B.19) is equivalent to the force-displacement relations for a
uniform beam that are familiar from classical structural analysis.

Member end-releases

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moment. That process of assigning a zero value to one term in the stiffness
matrix will affect the others terms because equilibrium has to be
maintained.

For a beam element, the six equilibrium equations in the local reference
system can be written as

/#=V 9 (B-20)

If one end of the member has a hinge, or other type of release that causes
the corresponding force to be equal to zero, Equation (B.20) requires
modification. A typical equation is of the following form:

j =1

(B. 21 )

If we know a specific value of f n is zero because of a release, the
corresponding displacement u n can be written as

7i—i l- 12 k

u =Y—u-+ T -^H.+r

n Z_-/ 7 1 Z_-/ Is J n

j =1 %

j=n +1 ^nn

(B. 22 )

Therefore, by substitution of Equation (B.22) into the other five
equilibrium equations, the unknown u n can be eliminated and the
corresponding row and column set to zero, or

f ij = k ij U ij + r v

The terms f n = 0 and the new stiffness terms are equal to

kij

k v

(B. 23 )

(B. 24 )

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There is a special case when the load is applied at the end release node. In
this case, the load must be altered to maintain the o moment transfer. This
special case is discussed in Appendix H.

B.4 Element mass matrix

The mass influence coefficient m t j for a structure is the force in the i th DOF
due to unit acceleration in thej f,! DOF. Applying this definition to a beam
element with distributed mass m (x) and using the principle of virtual
displacement, a general equation for can be derived

m ij= f m (x)Vi(x)Wj(x)dx (B.25)

The symmetric form of this equation shows that the mass matrix is
symmetric; = m ;i . If we use the same interpolation functions of

Equations (B.n) to (B. 16 ) as were used to derive the element stiffness
matrix into Equation (B. 23 ), the result obtained is known as the consistent
mass matrix. The integrals of Equation (B. 23 ) are evaluated numerically
or analytically depending on the function m(x). For an element with
uniform mass per unit length (i.e., m(x) = m), the integrals can be
evaluated analytically to obtain the element (consistent) mass matrix as

m e = mL

156

22 L

— 13L

420

22 L

4L 2

13L

-3L 2

420

13L

156

-22 L

420

— 13L

—3L 2

-22 L

4L 2

420

420 .

(B.26)

B.5 Element (applied) force vector

If the external forces p;(t), i = 1,2,3,4 ,5 and 6 are applied along the six
DOF at the two nodes of the finite element, the element force vector can be

172

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173

written directly. On the other hand, if the external forces are concentrated
forces p at locations Xj, the nodal force in the i th DOF is

P l (t) = Yj>'jV,(x,) (B-27)

This equation can be obtained by the principle of virtual displacement. If
the same interpolation functions of Equations (B.n) to (B. 16 ) are used to
derive the element stiffness matrix as used here, the results obtained are
called consistent nodal forces.

B.7 Nonlinear force-deflection relationship for the springs supports

The Impact_Deck software has the capability to calculate the response of
spring supports if the springs develop plastic behavior in the force-
displacement relationship. The spring can be considered as linear if the load
in the spring is below the elastic displacement Seias and the elastic force Feias
as shown in Figure B. 3 . If the load is reduced and the force-displacement is
below point 1 , the unload process returns along the same path as the loading
phase. The loading phase is shown using green arrows and the unloading
phase is shown using red arrows. However, if the load is greater than the
elastic displacement and it is in the loading stage, it follows the green
arrows until reaching the maximum force-displacement, point 2 . If the
unload occurs from this point, it will unload following a slope specified by
the user. In this case, the slope proceeds from point 2 to point 4 . If the force
never increases to point 2 again, the force-displacement will remain along
the line from point 2 to point 4 . If the force decreases to point 4 , zero force
is reached with a plastic permanent deflection. If the load increases again
until point 2 is reached, the original backbone curve is rejoined, proceeding
from point 2 towards point 3 . If the force reaches a maximum on the line
between point 2 and 3 and starts to decrease again, the load-deflection will
follow the same unload slope as the slope from point 2 to point 4 , but
starting from the new maximum force-deflection. If the force-deflection is
greater than point 3 , Impact_Deck assigns a zero value to this spring
because the maximum value was reached and failure occurs.

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Figure B.3 Force-displacement relation of the spring
support.

B.8 Transformation of the stiffness matrix from local to global
coordinate system

The Impact_Deck computer program has the option to calculate the
dynamic response of a flexible wall system such as the McAlpine
alternative flexible wall system when subjected to an impact load. The
McAlpine alternative flexible wall consists of a series of elastic beams
supported over a series of clustered pile groups. The beams transfer the
axial and shear forces to the pile cap by means of a shear key. Due to the
fact that the beams are not directly connected between the impact beams,
that is, they are connected by means of the cap beam shear key, the
mathematical model has to include a rigid beam element to model the
distance, stiffness, and mass of the shear key. Assuming that a barge train
moving at 3 ft/sec impacts the flexible wall at the end of a beam, and with
a time history duration of 3 sec, the whole impact process will have 9 ft of
length in contact, which occurs inside the shear key model length (i.e.,
contact solely with the concrete cap to the three-pile group) and not over
the beams. A typical McAlpine alternative flexible wall is presented in
Figure B. 4 , and this arrangement of the structural system can be observed.

Impact_Deck also models the effect of a shear key and pile cap along two
consecutive beams. Two rigid beam elements are used to model the end¬
points of the two incoming beams to the center of the pile cap along the
longitudinal center-line of the beams. However, the triangular arrangement

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Figure B.4 Typical McAlpine flexible wall system.

Cross-section

28 '

12 ' 2 " 8 ' 3 " 6 ' 7 "

t t

Barge

J lmpact

McAlpine Alternative
Flexible Wall

0> C

L8'

Barge Impact ab | > ! < [j

Figure B.5 McAlpine flexible wall mathematical model.

of piles allows the introduction of a possible rotation of the pile cap. That
rotation occurs at the center of rigidity of the pile cap with respect to its
supporting piles. In this case, the center of rigidity is at the center of the
pile cap, but along the transverse direction (perpendicular to the center-
line of the beams). To see the calculation of the center of rigidity, please
refer to Appendix E. The correct position of the equivalent translational
and rotational spring is at this center of rigidity, which does not coincide
with the longitudinal global axis of the beams. To connect the springs
located at the center of rigidity to the flexible beam elements and rigid
beam element (shear key), an additional rigid beam element is included

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perpendicular to the longitudinal beam (in the transverse direction) and
connected to a node at the center of rigidity of the pile cap. This concept
can be visualized in Figure B. 5 .

This rigid beam element has the same stiffness matrix in local coordinates
as a general beam element which stiffness matrix is

12EI

6 EI

12EI

6 EI

L 3

L 2

L 3

L 2

6 El

L 2

4 El

6 EI

L 2

2EI

12EI

6 EI

12EI

6 El

L 3

L 2

L 3

L 2

6 EI

2EI

6 EI

4EI

L 2

If the local axis of the element does not coincide with the global axis of the
transformation of axis has to be done to assemble the global stiffness
matrix of the structure. The transformation from local coordinate system
to global coordinate system, as shown in Figure B .5 can be done by the
transformation matrix.

First, let the equilibrium equations in local coordinates be,

{f'} = [k']{A} (B.29)

If each side of Equation (B. 29 ) is pre-multipled by transformation matrix,
it results in,

[ r ]{F} = [^][r]{A}

(B.30)

If Equation (B. 30 ) is again pre-multipled by the transpose of the
transformation matrix,

[ r f[r]{F} = [r] r [r][r]{A}

(B.31)

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And noting that the transformation matrix is orthonormal, that is,

[rf [r]=[z]

(B.32)

I]{F} = {F}

(B.33)

then,

{F} = [ r f[X'][r]{A} (B.34)

Equation (B. 34 ) relates the forces in global coordinates to the stiffness
matrix in local coordinates and the displacements in global coordinates.
Equation (B. 34 ) can be expressed as

{F} = \K\{ A} (B.35)

where [K] is the global stiffness matrix of the element. Finally, the stiffness
matrix in global coordinates can be expressed as multiplication of three
matrices, the transpose of the transformation matrix by the stiffness
matrix in local coordinates, and by the transformation matrix as

[AT] = [rf [k'\[T] (B.36)

It can be demonstrated by the equilibrium equations that the
transformation matrix has the form

cos0 sinQ 0 0 0 0

-sm0 cos0 0 0 0 0

0 0 1 0 0 0

0 0 0 cos0 sinQ 0

0 0 0 -sm0 cos0 0

0 0 0 0 0 1

The Impact_Deck computer program performs this matrix calculation to
transform the stiffness matrix in local coordinate system to global
coordinate system for the rigid beam element which is perpendicular to
the beam alignment as shown in Figure B. 6 . In that case, Equation (B. 8 )
uses a value of 9 = 90 degrees (the angle between the local axis to the

(B.37)

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global axis). The same transformation procedure is done for the element
mass matrix to transform from local to global coordinate system for the
perpendicular rigid beam element.

Figure B.6 Transformation of beam element coordinate
system (Local-Global).

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Appendix C: Guard wall - Equations of Motion
for the Mathematical Model

In structural dynamics the mathematical model of bodies of finite
dimensions undergoing translatory motion are governed by Newton’s
Second Law of Motion, expressed as

= m*a (1.1 bis)

at each time step t during motion. In the mathematical model of
transverse vibration, the forces acting on the flexible impact beam mass at
each time step t include (l) the impact force at time step t, ( 2 ) the elastic
restoring forces (of the beam), and ( 3 ) the damping forces (of the beam).
The mathematical model of the beam in the engineering formulation
described in this has a finite number of degrees of freedom (DOF) because
it is discretized using the finite element formulation. The engineering
formulations of equations of motion are solved using a numerical solution
method to determine the displacement and response forces at each pile
bent support feature. Each group of clustered piles is modeled as
transverse and longitudinal elastic/plastic springs. This is a dynamic
process, with the response forces responding to an impact pulse force time
history and each calculation of the equations of motion occurring at each
time step. The applied impact force can be given a constant velocity
parallel to the approach wall so that it is changing position with time. The
responses of the elastic beam over elastic or plastic springs can be
obtained by the use of the multiple degrees of freedom model (MDOF) and
the finite element formulation of the beam element. Impact_Deck includes
the damping forces by using the Rayleigh damping model. The response is
obtained using either the HHT-a method (Appendix F) or the well-known
Wilson-0 method (Appendix G). By using the Newton’s Second law and
applying it to a MDOF system, the resulting equations of motion can be
expressed as

mwoj+iciwoj+mmom^o} (2.1 bis)

where:

[M] = global mass matrix
[C] = global damping matrix

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[K] = global stiffness matrix
{F(t)> = global vector of external forces and moments
{u(t)}, {u(t)}, {u(t)} = relative acceleration, relative velocity, and relative
displacement for each DOF.

This appendix discusses the various relationships used in this engineering
methodology that are implemented in Impact_Deck.

C.l Element degrees of freedom (DOF) and interpolation functions

The longitudinal displacement (i.e., axial direction) of the beam element is
related to its two DOFs:

U ( X > t ) = Yj U i(. t W X ) (C- 1 )

1 = 1

where the function defines the displacement of the element due to

unit displacement u i: while constraining other DOF to zero. Thus <pi (x)
satisfies the following boundary conditions and are shown in Figure C.l,

i= 1 -Ti(0) = l, W 1 {L) = 0

(C. 2 )

i= 2 ;ip 2 (0) = 0, ip 2 (L)=l

(C. 3 )

Note that these DOF subscript values i = 1 and 2 correspond to Figure C.l
DOF subscripts i = 1 and 4.

The transverse displacement and rotation of the beam element is related
to its four DOF,

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Figure C.l Shape function for axial displacement
effect.

u(x,t) = Y^i(t)Wi(x) i = 2,3,5,6 (C.4)

where the function i/>;(x) defines the displacement of the element due to
unit displacement u it while constraining other DOF to zero. Thus, ipi(x)
satisfies the following boundary conditions,

i= i :tp 1 ('0J = l,\p' 1 (0)=\p 1 (L)=\p' 1 (L) = 0

i= 2 ty' 2 (0) — l,\p 2 (0) =ip 2 (L) = ip 2 (L) = 0

i= 3 : rp 3 (L) = l,ip 3 (0) = ip 3 (0) = ip 3 (L) = 0

i= 4 • ty\(L) = l,ip 4 (°) =ip' 4 (0) =\p 4 (L) = 0

Note that these DOF subscript values z = l, 2, 3, and 4 correspond to
Figure C.2 DOF subscripts i = 2, 3, 5, and 6.

(C. 5 )

(C.6)

(C. 7 )

(C.8)

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as illustrated next for the transverse displacement and rotation. Neglecting
shear deformations, the equilibrium equation for a beam loaded only at its
ends is

(C. 9 )

The general solution of Equation (C.9) for a uniform beam is a cubic
polynomial

+ flo

/ \

+ CL a

' x'

H-

(C. 10 )

The constants a t can be determined for each of the four sets of boundary
conditions of Equations (C.5) to (C.8), to obtain

( -vO

/ \

1-3

+ 2

(C.ll)

W 2 { x ) = l

( \

— 2 L

+ L

(C. 12 )

W 3 (*) = 3

/ \

— z

(C. 13 )

+ L

'x x3

(C. 14 )

These interpolation functions, illustrated in Figure C.2, can be used in
formulating the element matrices. The same process can be done with the
axial effect to obtain the basic interpolation functions,

Ti(*) = l —

(C. 15 )

t 2 (*) = -

(C.16)

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Figure C.2 Shape function for transverse
displacement and rotation effect.

C.2 Element stiffness matrix

Consider a beam element of length L with flexural rigidity EI(x). By
definition, the stiffness influence coefficient k t j of the beam element is the

force in the DOF i due to unit displacement in DOFj. Using the principle
of virtual displacement, the general equation for k t j, which is the stiffness

term for the transverse displacement and rotation, in the element stiffness
matrix is

(C. 17 )

The symmetric form of this equation shows that the element stiffness
matrix is symmetric; ky = kji- Equation (C.19) is a general result in the

sense that it is applicable to elements with arbitrary variation of flexural

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184

rigidity EI(x), although the interpolation functions of Equations (C.n) to
(C.14) are exact only for uniform elements. The associated errors can be
reduced to any desired degree by reducing the element size and increasing
the number of finite elements in the structural idealization. For a uniform
finite element with EI(x) = El, the integral of Equation (C.17) can be
evaluated analytically for i,j = 2,3,5, and 6, resulting in the
corresponding terms in the element stiffness matrix.

In the same way, for the axial contribution, the general equation for k t j,
which is the stiffness term for the axial displacement in the element
stiffness matrix is

= fEA(x)\v i (x)\v' j (x)dx (C.18)

For a uniform finite element with EA(x) = EA, the integral of Equation
(C.18) can be evaluated analytically for i,j = 1,4 resulting in the corres¬
ponding terms in the element stiffness matrix. Finally, when all terms are
calculated, the element stiffness matrix can be obtained as

12 EI

6EI

12 EI

6EI

L 3

L 2

L 3

L 2

*1=

6EI

L 2

4 El

6EI

L 2

2 EI

12 EI

6EI

12 El

6 El

L 3

L 2

L 3

L 2

6EI

2 EI

6EI

4 EI

L 2

(C.19)

These stiffness coefficients are the exact values for a uniform beam,
neglecting shear deformation, because the interpolation functions are the
true deflection shapes for this case. Observe that the stiffness matrix of
Equation (C.19) is equivalent to the force-displacement relations for a
uniform beam that are familiar from classical structural analysis.

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C.3 Member end-releases

When a structure has an internal pin (i.e., no moment transfer from one
element to the adjacent element), the DOF associated to the rotation must
have a stiffness value of zero for that DOF. In that way the element will
keep the ability to transfer the axial and shear force but not the bending
moment. That process of assigning a zero value to one term in the stiffness
matrix will affect the others terms because equilibrium has to be
maintained.

For a beam element, the six equilibrium equations in the local reference
system can be written as

4 =^-U.. (C. 20 )

If one end of the member has a hinge, or other type of release that causes
the corresponding force to be equal to zero, Equation (C.20) requires
modification. A typical equation is of the following form:

fn=T,KjUj

j=1

(C. 21 )

If we know a specific value of f n is zero because of a release, the
corresponding displacement u n can be written as

n—1 b 12 b

= y\L u + y -2L

j=1 ^nn j=n +1 %n

U J+ r n

(C. 22 )

Therefore, by substitution of Equation (C.21) into the other five
equilibrium equations, the unknown u n can be eliminated and the
corresponding row and column set to zero, or

fij = kij Uy +/’//

The terms f n = 0 and the new stiffness terms are equal to

kij

k..

(C. 23 )

(C. 24 )

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186

There is a special case when the load is applied at the end release node. In
this case, the load must be altered to maintain the o moment transfer. This
special case is discussed in Appendix H.

C.4 Element mass matrix

m ij= J m { x )Wi{x)Wj{x)dx (C. 25 )

The symmetric form of this equation shows that the mass matrix is
symmetric; j. If we use the same interpolation functions of

Equations (C.n) to (C.16) as were used to derive the element stiffness
matrix into Equation (C.20), the result obtained is known as the consistent
mass matrix. The integrals of Equation (C.20) are evaluated numerically or
analytically depending on the function m(x). For an element with uniform
mass per unit length (i.e., m(x) = m), the integrals can be evaluated
analytically to obtain the element (consistent) mass matrix as

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187

m e — mL

156

22 L

— 13L

420

22 L

4L 2

13L

420

13L

156

-22 L

420

— 13L

-22 L

4L 2

420

420 .

(C.26)

C.5 Element (applied) force vector

If the external forces p;(t), i = 1 , 2 , 3 , 4 , 5 , and 6 are applied along the six
DOF at the two nodes of the finite element, the element force vector can be
written directly. On the other hand, if the external forces are concentrated
forces at locations Xj, the nodal force in the i th DOF

Pi( t ) = YfjVi( x j) (C ‘ 27)

This equation can be obtained by the principle of virtual displacement. If
the same interpolation functions of Equations (C.n) to (C.i6) are used to
derive the element stiffness matrix as used here, the results obtained are
called consistent nodal forces.

C.6 Nonlinear force-deflection relationship for the springs supports

Impact_Deck has the capability to calculate the response of spring
supports if the springs develop plastic behavior in the force-displacement
relationship. The spring can be considered as linear if the load in the
spring is below the elastic displacement Seias and the elastic force Feias as
shown in Figure C.3. If the load is reduced and the force-displacement is
below point 1, the unload returns along the same path as the loading
phase. The loading phase is shown using green arrows and the unloading
phase is shown using red arrows. However, if the load is greater than the
elastic displacement and it is in the loading stage, it follows the green
arrows until reaching the maximum force-displacement, point 2. If the

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188

unload occurs from this point, it will unload following a slope specified by
the user. In this case, the slope proceeds from point 2 to point 4. If the
force never increases to point 2 again, the force-displacement will remain
along line from point 2 to point 4 until zero force is reached with a plastic
permanent deflection. If the load increases again until point 2 is reached,
the original backbone curve is rejoined proceeding from point 2 towards
point 3. If the force reaches a maximum on the line between point 2 and 3
and starts to decrease again, the load-deflection will follow the same
unload slope as the slope from point 2 to point 4, but starting from the
new maximum force-deflection. If the force-deflection is greater than
point 3, Impact_Deck assigns a zero value to this spring because the
maximum value was reached and failure occurs.

Figure C.3 Force-displacement relation of the spring
support.

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189

Appendix D Push-over analysis for batter-pile
bent system

D.l Bent geometry and analytical model used in the push-over
analysis

The pipe pile bent of Figure D.l is examined by push-over analysis
(Ebeling et al. 2012) to determine its load-displacement characteristics.
The Saul (1968), CPGA analytical model used in the analysis is shown in
Figure D.2. 1 The pipe pile bent is comprised of 24-inch-diameter,
concrete-filled pipe piles. Load-displacement plots will be determined by
push-over analysis for pinned-head and fixed-head conditions.

Figure D.l Pipe pile approach wall.

1 Pile numbers are reported in Figure D.2.

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Figure D.2 CPGA analytical model.

8.0

D.2 Pipe pile properties

Pipe pile properties are presented below:

Diameter D p of concrete-filled pipe pile = 2.0 ft = 24 in.

Area Ap of concrete-filled pipe pile = 3.142 ft 2 = 452 in. 2

Moment of inertia I P of concrete-filled pipe pile = 0.785 ft4 = 16300 in.4

Radius of gyration = 0.50 ft = 6 in.

Distance from neutral axis to extreme fiber c = 1.0 ft. = 12 in.

Modulus of elasticity E c = 504000 ksf = 3500 ksi

A simple interaction (axial load - moment) diagram is developed to help
in assessing the conditions where piles reach their moment or axial load
limits. The interaction diagram is based on the ultimate capacity of the pile
members. The procedures described in Rangan and Joyce (1992) can be
used to develop a simple interaction diagram for a concrete-filled pipe pile.
The interaction diagram points are

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191

• Pure axial compression

• Balance condition (axial compression and bending)

• Pure bending

• Pure axial tension

The value for pure axial compression is based only on the compressive
strength of the concrete. The 0.375-inch thick steel pipe casing was not
included in this calculation. The value for pure axial tension is based only
on the tensile strength of the steel pipe. Balance point and pure moment
conditions assumes the contribution only of the concrete in compression
on the compressive side of the neutral axis and contribution only of the
steel in tension on the tensile side of the neutral axis.

The interaction diagram for a 24-inch-diameter, concrete-filled pipe pile is
presented in Figure D.3.

Figure D.3 Simple interaction diagram for 24-inch-diameter pipe pile.

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192

The interaction diagram assumes that the pipe piles in axial compression
fail as a result of the materials (i.e., concrete and steel) reaching their
ultimate capacities, rather than by buckling. However buckling
computations will be needed to assure that this is the case. If buckling
loads are less than the ultimate axial compressive loads predicted by the
interaction diagram, then the buckling loads are to be used in the push¬
over analysis.

Piles are generally founded in soils that will not allow them to develop
their ultimate capacities. It is up to the engineer performing the push-over
analysis to consider axial load limitations imposed by the foundation
materials. This example also considers pile axial capacities that are limited
by side friction and tip resistance provided by the soil foundation.

D.3 Soil properties

A sand foundation with a stiffness that varies linearly with depth is
investigated. The coefficient of subgrade reaction (n/0 is assumed to be
50 pci. This value for m is corresponds to Terzaghi’s (1955) “recommended”
value for a moist medium-dense sand and within the scatter considering m
values cited in other technical literature (e.g., Davisson 1970). There is no
water table present in this case (i.e., a “dry” site).

The axial capacity based on soil limitations are 250 kips for piles in tension
and 1000 kips for piles in compression.

Relative stiffness factor (T):

T =

= 5.4 ft

D.4 Buckling evaluation

Buckling loads for the concrete-filled pipe piles are determined using
methods described in Yang (1996). Figures D.4 and D.5, after Figures 3
and 9 of Yang (1966), are provided for use in the analysis.

The coefficient of free standing length m is equal to the free standing
length L 0 divided by the relative stiffness factor T, or m = L 0 + T = 24 -r 5.4
= 4 - 45 ■

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193

Figure D.4 (After Figure 3 Yang 1966) Coefficient of critical
buckling strength.

Figure D.5 (After Figure 9 Yang 1966) Coefficient decrement of
buckling strength.

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194

The critical buckling load assuming no translation can be determined
using Figure D.4 (after Figure 3 in Yang 1966).

For pinned-top non-translating pile cap for Figure D.4, the coefficient of
critical buckling strength [G] is equal to 0.026 and the Euler critical
buckling load is

n El

PcR = ^t JL {G} = 3847 kips

For fixed-top non-translating pile cap from Figure D.4, the coefficient of
critical buckling strength [G] is equal to 0.056 and the Euler critical
buckling load is

■ CR

n El r .

—p-[G] = 7509 kips

The critical buckling load with translation can be determined using
Figure D.5 (after Figure 9 of Yang 1966). Entering Figure D.5 with a
coefficient of free standing length (m) equal to 4.45, the coefficient of
translation [Gr] is approximately equal to 0.21 for both pinned-head and
fixed-head piles. The critical buckling load assuming a translation ( Pcra ) is

p = p

1 CRA 1 CR

- 1

r C ^
2

\r )

It is desirable for the push-over analysis to have the critical buckling load
for various lateral displacements when performing a push-over analysis.
This has been accomplished with the aid of MathCAD (1998) for pinned-
head piles (Table D.i) and fixed-head piles (Table D.2).

D.5 Push-over analysis for pinned-head condition - dry site

The first push-over analysis is performed for a pinned-head condition at a
dry site using Saul’s (1968) method and the CPGA software. The results
are summarized in this section. The first incremental analysis was run
using an axial stiffness modifier ( C33 ) for the embedded portion of the pile
with a value of 1.00 for compression piles and 0.50 for tension piles
where:

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195

Where:

A =

This assumes that the pile is supported at its tip with all axial load (P)
transferred to the tip and:

8 = actual displacement of the pile under axial load (P)

Table D.l Euler critical buckling load - translating
pile top - pinned head condition

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196

Table D.2 Euler critical buckling load - translating pile top
- fixed head condition.

Euler critical buckling load - Translating pile top

G t :=0.21 Figure 9, Reference 1
Fixed top

r = 0.5 °ft

c := 1.0-ft

A := 3-in, 4-in.. 14-in

crA

(A):=P cr . l-G " A j

A ^A ) :=A

P crA( A ) =

°kip

5.932-103

5.407-10 3

4.881-10 3

4.355-103

3.83-103

3.304-103

2.778-10 3

2.253-10 3

1.727-10 3

1.202-10 3

675.846

150.188

This is a crude approximation of axial stiffness used to illustrate the push¬
over method. The actual value of the stiffness modifier (C33) should be
determined by appropriate analytical t-z models and/or pile load tests. To
obtain the maximum moment below the mudline it is necessary to include
a PMAXMOM data line where:

PMAXMOM =

And [H P ], the coefficient of horizontal load for pinned-head conditions, is
obtained from Figure D.6 (after Figure 7 of Yang 1966) and is equal to

0.20.

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197

Figure D.6, (After Figure 7 Yang 1966) Coefficient of horizontal load

capacity.

Therefore:

PMAXMOM -

5 4

— = 21 ft = 324 in.
0.20

The “ALLOW” and UNSP” data lines for CPGA do not represent actual
allowable loads and buckling loads but were included only to obtain pile
force and displacement results.

For the first increment of lateral loading, a trial and error process is used
to determine the lateral load driving Pile #3 to its axial tensile capacity of
250 kips. This is accomplished with a lateral load of 88 kips. The CPGA
input and output for this loading increment is presented below.

ERDC/ITL TR-16-1

CPGA INPUT FOR RUN #1 Pinned-head piles

10 BATTER PILE BENT PINNED TOP FILE:BP4

15 PROP 3834. 16300. 16300. 453. 0.5 0.0 1 3

20 PROP 3834. 16300. 16300. 453. 1.0 0.0 2

30 SOIL NH .050 L 72. 24. 1 TO 3

40 PIN 1 TO 3

50 ALLOW R 1000. 242. 1485. 933. 8544. 8544. 1 TO 3

70 UNSP S 0.6 0.6 500. 500. N 1 TO 3

80 PMAXMOM 324. 324. 1 TO 3

CPGA LOAD-DISPLACEMENT OUTPUT FOR RUN #1 Pinned-head piles

PILE CAP DISPLACEMENTS

LOAD

CASE DX DZ R

IN IN RAD

1 .307 9E + 01 -.3695E-01 .5633E-02

Horizontal
displacement
equals 3.1 inch

PILE FORCES IN LOCAL GEOMETRY

Ml & M2 NOT AT PILE HEAD FOR PINNED PILES

PILE FI F2 F3 Ml M2

Maximum moments
below the mudline

K K K IN-K IN-K

1 Pile #3 reaches its axial load capacity of 250 kips

ERDC/ITL TR-16-1

199

By trial-and-error push-over investigations, it can be shown that the next
failure mechanism will occur due to flexural yielding of the piles below the
mudline. This will be followed by buckling of Pile 2 followed by buckling of
Pile 1 as load is shifted from Pile 2 to Pile 1. In the next CPGA run, a low
stiffness modifier ( C 33 = 0.0001) is given to Pile 3 to eliminate its ability to
attract axial load. This amounts to releasing Pile 3 in its axial direction
since it has reached 250 kips of axial tensile capacity.

The CPGA analysis is performed with a final incremental barge impact
load of 54 kips producing a total axial load in Pile 2 of: 506.4 + 14.2 =

520.6 kips (compression). This is less than the 1000 kips axial
compressive capacity due to skin friction and end bearing. The additional
barge impact load of 54 kips brings the flexural demand on the piles below
the mudline to their yield capacities. Referring to the Figure D.3
interaction diagram for a compressive axial load of 500 kips the flexural
yield capacity of the piling is 700 ft-k (8400 in.-k) and the flexural demand
on Pile #2 is 2817 in.-k + 5524 in.-k = 8341 in.-k. This occurs at a total
lateral displacement of3.i + 6.o = 9.iin. Referring to previous buckling
calculations, an axial compressive load of 558 kip with 12.0 in. of lateral
displacement will induce buckling. Therefore, when subjected to an
additional 3 in. of lateral displacement, buckling of Pile #2 is expected to
be followed by buckling of Pile #3. It should be noted that CPGA does not
have the capability to introduce below mudline flexural hinges and
therefore it will be assumed for the purpose of constructing the load-
deformation curve that the stiffness of the system remains unchanged
between the points below mudline where flexural hinging develops and
buckling takes place.

A load-displacement plot for the pinned-head bent (solid blue curve) at a
dry site is presented in Figure D.8 that is located at the end of this
Appendix. This figure summarizes the resulting load displacement curves
from this and three other CPGA analyses that will be discussed
subsequently for pinned-head and fixed-head conditions at dry and wet
sand sites. These other three push-over analyses will be summarized prior
to discussing the resulting push-over load-displacement curves so that
comparisons can be made among the results four analyses.

ERDC/ITL TR-16-1

200

CPGA INPUT FOR RUN #2 Pinned-head piles

10 CANTILEVER BATTER PILE BENT FILE:BP5

15 PROP 3834. 16300. 16300. 453. 0.0001 0.0 3

20 PROP 3834. 16300. 16300. 453. 1.0 0.0 1 2

30 SOIL NH .047 L 72. 24. 1 TO 3

40 PIN 1 TO 3

50 ALLOW R 1000. 242. 1485. 933. 8544. 8544. 1 TO 3

70 UNSP S 0.6 0.6 500. 500. N 1 TO 3

80 PMAXMOM 324. 324. 1 TO 14

CPGA LOAD-DISPLACEMENT OUTPUT FOR RUN #2 Pinned-head piles

PILE CAP DISPLACEMENTS

LOAD

CASE DX DZ R

IN IN RAD

1 .5986E+01 .9708E-01 .1889E-01

PILE FORCES IN LOCAL GEOMETRY

PILE FI F2 F3 Ml M2

K K K IN-K IN-K

D.6 Push-over analysis for fixed-head condition - dry site

This section summarizes a second push-over analysis conducted for a
fixed-head condition at a dry site using the CPGA software. The first
incremental analysis was run using an axial stiffness modifier ( C 33 ) for the

ERDC/ITL TR-16-1

201

embedded portion of the pile with a value of l.oo for compression piles
and 0.50 for tension piles. This is a crude approximation of axial stiffness
used to illustrate the push-over method. The actual value of the stiffness
modifier (C33) should be determined by appropriate analytical t-z models
and/or pile load tests. To obtain the maximum moment below the mudline
it is necessary to include a FUNSMOM data line where:

FUNSMOM =

T Ln + ClT

Where [H/\, the coefficient of horizontal load for fixed-head conditions, is
obtained from Figure D.6 (after Figure 7 of Yang 1966) and is equal to
0.47, and

Lo = free standing length = 24 ft
a = coefficient of effective embedment obtained from Figure D.7
for “fixed top translating” = 1.7

Therefore,

FUNSMOM =

T L n +aT

UX—

5.4 24 + 1.7(5.4)

(X47 + 2

= 28.1 ft = 337 in.

As before, the “ALLOW” and UNSP” data lines for CPGA do not represent
actual allowable loads and buckling loads but were included only to obtain
pile force and displacement results.

For the first increment of lateral loading, a trial-and-error process is used
to determine the lateral load causing the pile to reach their moment
capacities at the pile to pile cap connection. This is accomplished with a
lateral load of 180 kips. CPGA input and output for this loading increment
is presented below.

ERDC/ITL TR-16-1

202

Figure D.7 (After Figure 2 Yang 1966) Effective embedment of pile

at buckling.

CPGA INPUT FOR RUN #1 Fixed-head piles

10 BATTER PILE BENT FIXED TOP FILE:BF4

15 PROP 3834. 16300. 16300. 453. 0.5 0.0 1 3

20 PROP 3834. 16300. 16300. 453. 1.0 0.0 2

30 SOIL NH .050 L 72. 24. 1 2 3

40 FIX 1 TO 3

50 ALLOW R 1000. 242. 1485. 933. 8544. 8544. 123

70 UNSP S 0.6 0.6 1000. 1000. N 1 2 3

80 FUNSMOM 337. 337. 1 3

ERDC/ITL TR-16-1

203

CPGA LOAD-DISPLACEMENT OUTPUT FOR RUN #1

Fixed-head piles

PILE CAP DISPLACEMENTS

LOAD

CASE DX DY DZ RX RY RZ

IN IN IN RAD RAD RAD

1 .2309E+01 .0000E+00 -.1013E+00 .0000E+00 .

3155E-02 .0000E+00

PILE FORCES IN LOCAL GEOMETRY

PILE FI F2 F3 Ml M2

From Figure B-3

K K K IN-K IN-K

(interaction diagram) for a

1 32.1 .0 -122.1 .0 7002.7

tensile load of 122 kips

FUNSMOM .0 -3817.5

the moment capacity =

2 32.3 .0 411.6 .0 7046.6

610 ft-k (7330 in-k).

FUNSMOM .0 -3846.5

Therefore moment

3 33.0 .0 -63.2 .0 7188.6

demand * moment

FUNSMOM .0 -3940.4

capacity at pile to pile cap

connection.

For the second increment of lateral loading, a trial-and-error process is
used to determine the lateral load causing pile #3 to reach its tensile load
capacity (250 k). This occurs with a lateral load increase of 64 kips. The
pile-to-pile cap connection is changed from fix to pin to capture the
yielding that occurred in Run #1.

ERDC/ITL TR-16-1

204

CPGA LOAD-DISPLACEMENT OUTPUT FOR RUN #2 Fixed-head piles

PILE CAP DISPLACEMENTS

LOAD

CASE DX DZ R
IN IN RAD

1 .2121E + 01 .14 93E-01 .4192E-02

PILE FORCES IN LOCAL GEOMETRY
PILE FI F2 F3 Ml M2
K K K IN-K IN-K

1 5.9 .0 18.0 .0 -1920.8

2 6.0 .0 376.7 .0 -1937.50

3 6.2 .0 -186.0 .0 -2014.8

By trial-and-error push-over investigations, it can be seen that next failure
mechanism will occur due to flexural yielding of the piles below the
mudline. This will be followed by buckling of Pile 2 followed by buckling of

Pile #3 has a tensile load
of 63.2 k from Run #1 and
186.0 k from Run #2 giving
a total axial tensile load of
249.2 k « 250 k. tensile
capacity reached.

ERDC/ITL TR-16-1

205

Pile l as load is shifted from Pile 2 to Pile l. In the next CPGA run, a low
stiffness modifier ( C 33 = o.oooi) is given to Pile 3 to eliminate its ability to
attract axial load. This amounts to releasing Pile 3 in its axial direction
since it has reached 250 kips of axial tensile capacity.

The CPGA analysis is performed with a final incremental barge impact
load of 22 kips producing a total axial load in Pile 2 of 411.6 + 376.7 + 5.8
= 794.1 kip (compression). This is less than the 1000 kip axial compressive
capacity due to skin friction and end bearing. The additional barge impact
load of 22 kips brings the flexural demand on the piles below the mudline
to their yield capacities. Referring to the Figure D.3 interaction diagram
for a compressive axial load of 800 kips, the flexural yield capacity of the
piling is 670 ft-k (8040 in.-k) and the flexural demand on Pile #2 is
3846 in.-k + 1937 in.-k + 2246 in.-k = 8029 in.-k indicating demand is
approximately equal to capacity. This occurs at a total lateral displacement
of 2.3+ 2.1 + 2.4 = 6.8 in. Referring to previous buckling calculations, an
axial compressive load of 800 kips at about 12.5 in. of lateral displacement
will cause buckling. Therefore when subjected to an additional 6 in. of
lateral displacement, buckling of Pile #2 is expected to be followed by
buckling of Pile #3. It should be noted that CPGA does not have the
capability to introduce below-mudline flexural hinges; therefore, it will be
assumed for the purpose of constructing the load-deformation curve that
the stiffness of the system remains unchanged between the points below
mudline where flexural hinging develops and buckling takes place.

ERDC/ITL TR-16-1

CPGA INPUT FOR RUN #3 Fixed-head piles

10 BATTER PILE BENT FIXED TOP FILE:BF6
15 PROP 3834. 16300. 16300. 453. 1.0 0.0 1 2
20 PROP 3834. 16300. 16300. 453. 0.0001 0.0 3
30 SOIL NH .050 L 72. 24. 1 TO 3
40 PIN 1 TO 3

50 ALLOW R 1000. 242. 1485. 933. 8544. 8544. 1 TO 3

70 UNSP S 0.6 0.6 500. 500. N 1 TO 3

80 PMAXMOM 324. 324. 1 TO 3

90 BATTER 4. 2 TO 3

100 ANGLE 0.123

110 PILE 1 0. 0. 0.

120 PILE 2 7. 0. 0.

130 PILE 3 14. 0. 0.

140 LOAD 1 22. 0. 200. 0. 0. 0.

190 TOUT 1234567
200 FOUT 1234567
210 PFO 1 TO 3

CPGA OUTPUT FOR RUN #3 Fixed-head piles

PILE CAP DISPLACEMENTS

LOAD

CASE DX DZ R
IN IN RAD

1 .2407E+01 .9850E-01 .9300E-02

PILE FORCES IN LOCAL GEOMETRY
PILE FI F2 F3 Ml M2
K K K IN-K IN-K

1 6.7 .0 198.0 .0 -2179.2

2 6.9 .0 5.8 .0 -2245.7

3 7.4 .0 -.2 .0 -2398.8

A load-displacement plot for the fixed-head bent (solid green curve) at a
dry site is presented in Figure D.8.

ERDC/ITL TR-16-1

207

D.7 Submerged Site

A submerged sand foundation with a stiffness that varies linearly with
depth is investigated. The coefficient of subgrade reaction ( m ) is assumed
to be 30 pci. This value for m is corresponds to Terzaghi’s (1955)
“recommended” value for a submerged medium-dense sand per Table 3.2
of Section 3 in Ebeling, et al. (2012). This is sometimes referred to as a
“wet” site in that report.

The axial capacity based on soil limitations are 250 kips for piles in tension
and 1000 kips for piles in compression.

Relative stiffness factor (T):

T = 5

E J p

6.0

D.8 Buckling evaluation

The coefficient of free standing length m is equal to the free standing
length L 0 divided by the relative stiffness factor T, or m = Lo+ T = 24 + 6.0
= 4.0.

Assuming no translation, the critical buckling load can be determined
using Figure D.4 (after Figure 3 in Yang 1966).

For pinned-top non-translating pile cap from Figure D.4, the coefficient of
critical buckling strength [G] is equal to 0.030 and the Euler critical
buckling load is

71 E J P

■ CR

[G] = 3256 kips

For fixed-top non-translating pile cap from Figure D.4, the coefficient of
critical buckling strength [G] is equal to 0.062 and the Euler critical
buckling load is

ERDC/ITL TR-16-1

208

n 2 E c I r n

Pcr = -^[G] = 6728 kips

The critical buckling load with translation can be determined using
Figure D.5 (after Figure 9 of Yang 1966). Entering Figure D.5 with a
coefficient of free standing length (m) equal to 4.0, the coefficient of
translation [Gr] is approximately equal to 0.20 for both pinned-head and
fixed-head piles. Assuming a translation ( Pcra ), the critical buckling load is

P = P

1 CRA 1 CR

-1

f c)

<r J

ERDC/ITL TR-16-1

Table D.3 Euler critical buckling load - translating pile
top - pinned head condition.

ERDC/ITL TR-16-1

210

Table D.4 Euler critical buckling load - translating pile
top - fixed head condition.

Euler critical buckling load - Translating pile top

Gj :=0.20 Figure 9, Reference 1
Fixed top

0.5

0.5-ft

c := 1.0 ft

A : = 3in,4in.. 14 in

P crA^ A ) :-P cr'( 1_ G T'-

A X (A):=A

P crA ( A ) .yp A _ 2 ^ A ) .jn

5.383-103

4.934-10 3

4.486-10 3

4.037-10 3

3.588-103

3.14-10 3

2.691-103

2.243-10 3

1.794-103

1.346-10 3

897.115

448.557

D.9 Push-over analysis for pinned-head condition - wet site

This section summarizes a third push-over analysis conducted for a
pinned-head condition at a wet site using Saul’s (1968) method and the
CPGA software. The first incremental analysis was run using an axial
stiffness modifier (C 33 ) for the embedded portion of the pile with a value of
1.00 for compression piles and 0.50 for tension piles where:

Where:

ERDC/ITL TR-16-1

211

A =

This assumes that the pile is supported at its tip with all axial load (P)
transferred to the tip and:

8 = actual displacement of the pile under axial load (P)

This is a crude approximation of axial stiffness used to illustrate the push¬
over method. The actual value of the stiffness modifier (C33) should be
determined by appropriate analytical t-z models and/or pile load tests. To
obtain the maximum moment below the mudline, it is necessary to include
a PMAXMOM data line where:

PMAXMOM =

Where \H P ], the coefficient of horizontal load for pinned-head conditions,
is obtained from Figure D.6 (after Figure 7 of Yang 1966) and is equal to
0.21.

Therefore:

T 0

PMAXMOM = T —i = — = 28.57 ft = 343 in.

Kl °- 21

The “ALLOW” and UNSP” data lines for CPGA do not represent actual
allowable loads and buckling loads but were included only to obtain pile
force and displacement results.

For the first increment of lateral loading, a trial-and-error process is used
to determine the lateral load driving Pile #3 to its axial tensile capacity of
250 kips. This is accomplished with a lateral load of 88 kips. CPGA input
and output for this loading increment is presented below.

ERDC/ITL TR-16-1

CPGA INPUT

FOR

RUN #1 Pinned-head piles

10 BATTER PILE

BENT PINNED TOP FILE:BP8

15 PROP

3834 .

16300. 16300. 453. 0.5 0.0 1 3

20 PROP

3834.

16300. 16300. 453. 1.0 0.0 2

30 SOIL

.030

L 72. 24. 1 TO 3

40 PIN 1

50 ALLOW

1000.

242. 1485. 933. 8544. 8544. 1 TO 3

70 UNSP

S 0

. 6

0 .

6 500. 500. N 1 TO 3

80 PMAXMOM

343

343. 1 TO 3

90 BATTER 4

. 2

TO 3

100 ANGLE 0

. 1

110 PILE

0 .

120 PILE

7 .

0 .

130 PILE

14 .

. 0 .

140 LOAD

88.

. 200. 0. 0. 0.

190 TOUT

2 3

5 6 7

200 FOUT

2 3

5 6 7

210 PFO

PILE CAP DISPLACEMENTS

LOAD

CASE DX DZ R

Horizontal
displacement
equals 3.1 inch

IN IN. RAD

1 .30 96E+01 -.37 8 9E-01 .5660E-02

PILE FORCES IN LOCAL GEOMETRY

Maximum moments
below the mudline

LOAD CASE

ERDC/ITL TR-16-1

213

PILE FI F2 F3 Ml M2

K K K IN.-K IN.-K

1 7.9 .0 -45.7 .0 -2700.6

2 8.0 .0 508.5 .0 -2728.53

3 8.2 .0 -251.2 .0 -2829.1

Pile #3 reaches its axial load
capacity of 250 kips

~k ~k ~k ~k ~k ~k

By trial-and-error push-over investigations, it can be shown that the next
failure mechanism will occur due to flexural yielding of the piles below the
mudline. This will be followed by buckling of Pile 2 followed by buckling of
Pile 1 as load is shifted from Pile 2 to Pile 1. In the next CPGA run a low
stiffness modifier (C 33 = 0.0001) is given to Pile 3 to eliminate its ability to
attract axial load. This amounts to releasing Pile 3 in its axial direction
since it has reached 250 kips of axial tensile capacity.

The CPGA analysis is performed with a final incremental barge impact load
of 52 kips producing a total axial load in Pile 2 of 508.5 + 13.8 = 522.3 kips
(compression). This is less than the 1000 kips axial compressive capacity
due to skin friction and end bearing. The additional barge impact load of
52 kips brings the flexural demand on the piles below the mudline to their
yield capacities. Referring to the Figure D.3 interaction diagram for a
compressive axial load of 500 kips, the flexural yield capacity of the piling is
700 ft-k (8400 in.-k) and the flexural demand on Pile #2 is 2728.5 in.-k +
5630.7 in.-k = 8359.2 in.-k. This occurs at a total lateral displacement of
3.1 + 6.3 = 9.4 in. Referring to previous buckling calculations, an axial
compressive load of 560 kip with 12.0 in. of lateral displacement will induce
buckling. Therefore, when subjected to an additional 3 in. of lateral
displacement, buckling of Pile #2 is expected to be followed by buckling of
Pile #3. It should be noted that CPGA does not have the capability to
introduce below mudline flexural hinges; therefore, it will be assumed for

ERDC/ITL TR-16-1

214

the purpose of constructing the load-deformation curve that the stiffness of
the system remains unchanged between the points below mudline where
flexural hinging develops and buckling takes place.

CPGA INPUT

FOR

RUN

#2 Pinned-head piles

CANTILEVER BATTER

PILE BENT FILE:BP5

PROP 3834. 16300.

16300.

453. 0.0001 0.0 3

PROP 3834. 16300.

16300.

453. 1.0 0.0 1 2

SOIL NH

.030

L 72.

24 . 1

TO 3

PIN 1 TO

ALLOW R

1000.

242 .

1485.

933. 8544. 8544. 1 TO 3

UNSP S 0

.6 0.

6 500

. 500.

N 1 TO 3

PMAXMOM

343.

1 TO 14

BATTER 4

. 2 TO 3

100

ANGLE 0

. 1 2

110

PILE 1

0 . 0 .

0 .

120

PILE 2

7. 0.

0 .

130

PILE 3

14. 0

. 0 .

140

LOAD 1

52. 0

. 200

. 0 . 0 .

0 .

190

TOUT 1

2 3 4

5 6

200

FOUT 1

2 3 4

5 6

210

PFO 1 TO 3

~k ~k ~k ~k ~k ~k

PILE CAP DISPLACEMENTS

LOAD

CASE DX DZ R

IN IN. RAD

1 .62 65E+01 .9716E-01 .1972E-01

ERDC/ITL TR-16-1

215

PILE FORCES IN LOCAL GEOMETRY

LOAD CASE - 1

PILE FI F2 F3 Ml M2

K K K IN.-K IN.-K

1 15.9 .0 195.3 .0 -5464.0

2 16.4 .0 13.8 .0 -5630.7

3 17.4 .0 -.5 .0 -5981.1

A load-displacement plot for the pinned-head bent (dashed blue curve) at
the wet site is presented in Figure D.8.

D.10 Push-over analysis for fixed-head condition - wet site

This section summarizes a fourth push-over analysis conducted for a
fixed-head condition at a wet site using Saul’s (1968) method and the
CPGA software. The first incremental analysis was run using an axial
stiffness modifier (C33) for the embedded portion of the pile with a value of
1.00 for compression piles and 0.50 for tension piles. This is a crude
approximation of axial stiffness used to illustrate the push-over method.
The actual value of the stiffness modifier (C33) should be determined by
appropriate analytical t-z models and/or pile load tests. To obtain the
maximum moment below the mudline, it is necessary to include a
FUNSMOM data line where:

FUNSMOM =

T Ln + ClT

Where \Hf\, the coefficient of horizontal load for fixed-head conditions, is
obtained from Figure D.6 (after Figure 7 of Yang 1966) and is equal to
0.52, and:

Lo = free standing length = 24 ft

ERDC/ITL TR-16-1

216

a = coefficient of effective embedment obtained from Figure D.y
for “fixed top translating” = 1.75

Therefore,

FUNSMOM =

Lq + uT
2

6.0 24 + 1.7(6.0)

052 + 2

= 28.6 ft = 344 in.

As before, the “ALLOW” and UNSP” data lines for CPGA do not represent
actual allowable loads and buckling loads but were included only to obtain
pile force and displacement results.

CPGA INPUT

FOR RUN

#1 Fixed-head piles

10 BATTER PILE

BENT

FIXED TOP FILE:BF7

15 PROP 3834.

16300.

16300. 453. 0.5 0.0 1 3

20 PROP 3834.

16300.

16300. 453. 1.0 0.0 2

30 SOIL NH

.030 L 72

. 24. 1 2 3

40 FIX 1 TO

50 ALLOW R

1000. 242

. 1485. 933. 8544. 8544. 123

70 UNSP S 0

. 6

0.6 1000. 1000. N 1 2 3

80 FUNSMOM

344

. 344 .

1 3

85 FUNSMOM

344

. 344.

90 BATTER 4

. 2

TO 3

100 ANGLE 0

. 1

2 3

110 PILE 1

0 .

0 . 0 .

120 PILE 2

7 .

0 . 0 .

130 PILE 3

14 .

0 . 0 .

140 LOAD 1

180

. 0. 200. 0. 0. 0.

190 TOUT 1

2 3

4 5 6

200 FOUT 1

2 3

4 5 6

210 PFO 1

ERDC/ITL TR-16-1

217

*****************************************************************

* * * * * *

PILE CAP DISPLACEMENTS

LOAD

CASE DX DY DZ

IN IN IN

1 .2462E+01 .0000E+00 -.1102E+00

From Figure B-3 (interaction
diagram) for a tensile load of
122 kips the moment capacity «
610 ft-k (7330 in-k). Therefore
moment demand « moment
capacity at pile to pile cap
connection.

FUNSMOM .0 -3699.8

2 31.5 .0 431.9 .0 7090.5

FUNSMOM .0 -3729.1

3 32.1 .0 -72.9 .0 7232.5

FUNSMOM .0 -3820.5

PILE FORCES IN LOCAL GEOMETRY

LOAD CASE - 1

PILE FI F2 F3 Ml M2

K K K IN.-K IN.-K

1 31.2 .0 -132.9 .0 7045.0

'k'k'k'k'k'k'k'k'k'k

ERDC/ITL TR-16-1

218

pile-to-pile cap connection is changed from fix to pin to capture the
yielding that occurred in Run #1.

CPGA INPUT

FOR

RUN #2 Fixed-head piles

BATTER PILE BENT FIXED TOP FILE:BF8

PROP 3834. 16300. 16300. 453. 0.5 0.0 1 3

PROP 3834. 16300. 16300. 453. 1.0 0.0 2

SOIL NH

.030

L 72. 24. 1 TO 3

PIN 1 TO

ALLOW R

1000.

242. 1485. 933. 8544. 8544. 1 TO 3

UNSP S 0

.6 0.

6 500. 500. N 1 TO 3

PMAXMOM

344 .

344. 1 TO 3

BATTER 4

. 2 TO 3

100

ANGLE 0

. 1 2

110

PILE 1

0 . 0 .

0 .

120

PILE 2

7. 0.

0 .

130

PILE 3

14 . 0

. 0 .

140

LOAD 1

70. 0

. 200. 0. 0. 0.

190

TOUT 1

2 3 4

5 6 7

200

FOUT 1

2 3 4

5 6 7

210

PFO 1 TO 3

~k ~k ~k ~k ~k ~k

PILE CAP DISPLACEMENTS

LOAD

CASE DX DZ R

IN IN. RAD

1 .2 4 37E+01 -.217 9E-02 .4668E-02

ERDC/ITL TR-16-1

219

PILE FORCES IN LOCAL GEOMETRY

LOAD CASE - 1

PILE FI F2 F3 Ml M2

K K K IN.-K IN.-K

1 6.2 .0 -2.6 .0 -2131.9

2 6.3 .0 419.3 .0 -2151.91

3 6.5 .0 -207.2 .0 -2235.1

Pile #3 has a tensile load
of 32.1 k from Run #1 and

207.2 k from Run #2 giving
a total axial tensile load of

239.3 k « 250 k. tensile
capacity reached.

~k ~k ~k ~k ~k ~k

By trial-and-error push-over investigations it can be seen that next failure
mechanism will occur due to flexural yielding of the piles below the
mudline. This will be followed by buckling of Pile 2 followed by buckling of
Pile l as the load is shifted from Pile 2 to Pile 1. In the next CPGA run, a
low stiffness modifier ( C33 = 0.0001) is given to Pile 3 to eliminate its
ability to attract axial load. This amounts to releasing Pile 3 in its axial
direction since it has reached 250 kips of axial tensile capacity.

The CPGA analysis is performed with a final incremental barge impact
load of 22 kips producing a total axial load in Pile 2 of 431.9 + 419.3 + 5.9
= 857.1 kips (compression). This is less than the 1000 kips axial
compressive capacity due to skin friction and bearing. The additional
barge impact load of 22 kips brings the flexural demand on the piles below
the mudline to their yield capacities. Referring to the Figure D.3
interaction diagram for a compressive axial load of 800 kips the flexural
yield capacity of the piling is 675 ft-k (8100 in. k) and the flexural demand
on Pile #2 is 3729 in. k + 2152 in. k + 2385 in. k = 8266 in. k which
indicates that the demand is slightly greater than the capacity. This occurs
at a total lateral displacement of 2.5 + 2.4 + 2.6 = 7.5 in. Referring to
previous buckling calculations an axial compressive load of 857 kips at
about 13 in. of lateral displacement will cause buckling. Therefore when
subjected to an additional 5.5 in. of lateral displacement, buckling of Pile
#2 is expected to be followed by buckling of Pile #3. It should be noted
that CPGA does not have the capability to introduce below mudline

ERDC/ITL TR-16-1

220

flexural hinges and therefore it will be assumed for the purpose of
constructing the load-deformation curve that the stiffness of the system
remains unchanged between the points below mudline where flexural
hinging develops and buckling takes place.

CPGA INPUT

FOR

RUN #3 Fixed-head piles

10 BATTER

PILE

BENT FIXED TOP FILE:BF6

15 PROP 3834.

16300. 16300. 453. 1.0 0.0 1 2

20 PROP 3834.

16300. 16300. 453. 0.0001 0.0 3

30 SOIL NH

.030

L 72. 24. 1 TO 3

40 PIN 1 TO 3

50 ALLOW R

1000.

242. 1485. 933. 8544. 8544. 1 TO 3

70 UNSP S

0.6

0 .

6 500. 500. N 1 TO 3

80 PMAXMOM

344

344. 1 TO 3

90 BATTER

4. 2

TO 3

100 ANGLE

0 . 1

110 PILE 1

0 .

120 PILE 2

7 .

0 .

130 PILE 3

14 .

. 0 .

140 LOAD 1

22 .

. 200. 0. 0. 0.

190 TOUT 1

2 3

5 6 7

200 FOUT 1

2 3

5 6 7

210 PFO 1

TO 3

*****************************************************************

******

PILE CAP DISPLACEMENTS

LOAD

CASE DX DZ R

IN IN RAD

1 .2 645E+01 .984 9E-01 .9010E-02

*****************************************************************

ERDC/ITL TR-16-1

221

PILE FORCES IN LOCAL GEOMETRY

LOAD CASE - 1

PILE FI F2 F3 Ml M2

K K K IN.-K IN.-K

1 6.7 .0 198.0 .0 -2314.0

2 6.9 .0 5.9 .0 -2384.5

3 7.4 .0 -.2 .0 -2545.1

A load-displacement plot for the fixed-head bent (dashed green curve) at
the wet site is presented in Figure D.8. The resulting structural system
versus displacement plots characterizes the potential energy capacity of
the particular batter pile bent being analyzed. The push-over results for
four systems are shown in this figure.

In Figure D-8 the load-displacement results by the Saul (1968) method for
a dry site (m = 50 pci) are represented by solid line and those for a
submerged or wet site (m = 30 pci) by dashed lines. Yang (1966) and
COM624G methods cannot be used for batter-pile systems because they
are only applicable to single vertical pile analysis. The methods can be
used for systems comprised of multiple vertical piles since a single pile
from the system can be analyzed and the load-displacement results for the
entire system derived based on the behavior of that single pile.

The load-displacement curves for the fixed-head pile system have four
break points designating places where pile or soil yielding occurs. The
number of yield points and the type of yielding will be pile bent and
foundation dependent. For the particular pile-bent-foundation system
investigated, the first break point (one with lowest displacement demand)
occurs when flexural yielding takes place at the pile-to-pile cap
connection. The second breakpoint occurs when Pile 3 yields in axial

ERDC/ITL TR-16-1

222

tension (a foundation to pile transfer mechanism). 1 The third breakpoint
occurs when flexural yielding takes place in the piles below the mudline.
The fourth breakpoint occurs when Pile 2 buckles. Pile buckling quickly
results in pile-bent system failure with little reserve potential energy
capacity in the system.

There is little difference between the behaviors of submerged (wet) sites
and dry sites, recognizing of course that lock approach wall bent systems
will always be submerged. With batter-pile systems, the resistance to

1 Recall pile numbers are reported in Figure D.2.

ERDC/ITL TR-16-1

223

lateral load comes principally from pile axial stiffness and not from
flexural stiffness, as is the case with vertical pile systems. Therefore,
changes in lateral subgrade resistance (e.g., m) have little effect on system
load-displacement behavior.

It can easily be recognized from Figure D-8 that the fixed-head system
(green curves) has much greater potential energy capacity than the free-
head system (blue curves). The free-head system does not possess the
added lateral force resistance provided by rigid pile-to-pile cap
connections (which is the first break point for the fixed-head system).

The information contained in this appendix illustrates the push-over
analysis for a pinned-head and fixed-head batter pile bent using the Corps
computer program CPGA (X0080) for wet and dry sites (i.e., m = 30 pci
and 50 pci, respectively). Note that potential failure mechanisms and the
sequence in which they form will likely be different for other batter-pile
bent system groups and pile configurations.

ERDC/ITL TR-16-1

224

Appendix E: Formulation for the rotational
spring stiffness for the McAlpine flexible
approach wall clustered group of vertical
piles model

E.l Calculation of the Center of Rigidity of a pile group for the
McAlpine flexible wall model

The McAlpine flexible wall system is supported over pile groups consisting
of three piles in a triangular arrangement as shown in Figure E.l.
Impact_Deck computer program has the capability to perform a dynamic
analysis of this structure that introduces an important change to the finite
element model from the other two types of approach wall (i.e., guard wall
and impact deck) structural configurations. This model change is the
inclusion of a rotational spring at the central rigid link, in addition to
moving the two translational elastic-plastic springs to the central rigid link.
This addition is needed because the piles in the group are not in line with
each other and a torsional resistance is provided by this configuration,
which means that the pile bent rotates about a Center of Rigidity (C.R.),
which is not in line with the wall beam. First, the location of the C.R.
calculation must be performed because the C.R. position identifies the
length of the rigid element which is perpendicular to the beam alignment.

Figure E.l. Plan view of the McAlpine flexible alternative approach wall system.

The length of the rigid element, which is perpendicular to the beam
elements, is calculated as the distance between C.R. and the central blue

ERDC/ITL TR-16-1

225

cross as shown in Figure E.2. Figure E.2 also shows the relation between
the pile cap local axis and the beam global axis. This Figure also
demonstrates that the alignment of the centerline of the beam does not
coincide with the location of the piles.

Figure E.2 Relation between Global-Axis and central support Local-Axis.

Figure E.3 shows the dimensions and the distances needed to calculate the
center of rigidity

Figure E.3 Location of the Center of Rigidity.

ERDC/ITL TR-16-1

226

The equations to calculate the C.R. are the following. First, the moment of
inertia for an element with a circular cross-sectional is

t ^ j4

I = — * a

(E.l)

The translational elastic stiffness of the pile fixed at bottom and top is

Kile =

12 * E * /

(E. 2 )

If the piles have the same cross-sectional area, same modulus of elasticity,
and same length then, following the notation presented in Figure E.3, the
coordinates of the Center of Rigidity are

v _ i=l

^CR r

E k n

i=1
n

I Z k xi*yi

T/ - _ i =1

1 CR — 7

T, k xi

i=1

(E. 3 )

Following the same notation, the equivalent translational spring stiffness
in the global coordinate system are calculated as

( k x)„=E k x> (E.4)

7=1

{ k r)«,=E k r, (E.5)

1=1

where n = number of piles . Finally, the length of the rigid element which
is perpendicular to the beam can be calculated as

T — X —p

^Rigid Link CR C A'

(E. 6 )

ERDC/ITL TR-16-1

227

E.2 Numerical example for the calculation of the Center of Rigidity

The definitions of the variables are presented in Figure E.3.

Data :

d = 5 ft 8 in. = pile diameter
f'c = 5,000 psi

E = 57,000*(5,000) 1/2 = 4,030,508.65 psi = 580,393.25 ksf = Modulus of elasticity
L = 20 ft = height of the pile above ground
ex= beam width/2 = 3 ft- 3.5 in.

Calculations :

I = n* (5.6666) 4 / 64 = 50.613 ft 4

kpile = 12* 580,393.25* 50.613/ (20) 3

= 44,063.16 kip /ft

kxi = k X 2 = k X 3 = 44,063.16 kip /ft

kyi = ky2 = ky3 = 44,063.16 kip /ft

xi = (28-18)/2 + 18 = 23ft

x 2 = (28-18)/2 = 5ft

x 3 = (28-18)/2 = 5ft

yi = 19.5/2 = 9.75 ft

y 2 = (19.5 -11.5)/2 +11.5 = 15.5 ft

y 3 = (19.5 -11.5)/2 = 4 ft

X C r = (44,063.16 *(23 + 5 + 5)) /132,189.5

ERDC/ITL TR-16-1

228

= lift

Y C r = (44,063.16 *(9.75+15.5+4))/132,189.5 = 9.75ft

(k x ) eq . = 132,189.5 kip /ft

(k Y )eq. = 132,189.5 kip/ft

l-Rigid Link = 11 - 3.291666 = 7.708333 ft

E.3 Calculation of the rotational spring stiffness of a pile group for
the McAlpine flexible wall model

The rotational stiffness is developed in a pile group when the response
forces are not aligned to the center of rigidity, forming moment arms from
the line of action of the forces and the center of rigidity. That concept is
presented in Figure E.4. Each one of the forces contributes to the resultant
moment around the axis normal to the plan view.

Figure E.4 Definition of the forces and
distances generated when the pile cap rotate.

The resultant moment creates a rotation in the structure. This angle of
rotation is associated to the ratio of the lateral displacement to the
distance from the center of each pile to the center of rigidity, as shown in
Figure E.5.

ERDC/ITL TR-16-1

229

Figure E.5 Rotational angle definition
when the pile cap rotate.

The governing equations to calculate the rotational spring stiffness are the
following. The distance from the line of action of the shear forces in the
piles to the center of rigidity are,

d 1 = x 1 -X CR (E.7)

(E.8)

The resultant moment (torsion) due to the shear forces in the piles is

i=l

(E.9)

where the forces in pile number one can be defined as the translational
stiffness times the displacement achieved,

Fi = kj * Sj ( E .1Q)

and the rotation of the pile cap can be calculated as

ERDC/ITL TR-16-1

230

this assumes that all piles will rotate the same amount (i.e., rigid body
motion). Following that concept, the lateral displacement and shear forces
at piles two and three are

S2 = 0 * d 2

(E.12)

F 2 =k 2 *S 2

(E.13)

-X-

- ^

(E.14)

F 3 =k 3 *S 3

(E.15)

Finally, the rotational spring stiffness can be calculated as the ratio of the
resultant moment divided by the angle of rotation, as

l c M C r

K ~ e

(E.16)

E.4 Numerical example of the calculation of the rotational spring
stiffness of a pile group for the McAlpine flexible wall model

Using the numerical results obtained in section E.2, the calculation of the
center of rigidity is presented next.

Data:

kxi = k X 2 = k X 3 = 44,063.16 kip /ft

kyi = ky2 = ky3 = 44,063.16 kip /ft

di = 23 -11 = 12 ft

d 2 = d 3 = (5.75 2 + 6 2 ) 1/2 = 8.310ft

B= 0.063 ft
Calculations:

Fi = 44,063.16 * 0.063 = 2,775.98 kips

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231

tan 0 = 0.063/12 = 0.00525

0 = arctan (0.00525) = 0.00525 (small angle)

d 2 = 0.00525 * 8.310 = 0.0436ft

F 2 = 44,063.16* 0.0436 = 1,921.15 kips

d 3 = 0.00525 * 8.310 = 0.0436 ft

F 3 = 44,063.16* 0.0436 = 1,921.15 kips

Mcr = 2,775.98* 12 + 1,921.15* 8.310

+ 1,921.15* 8.310 = 65,241.27 kip * ft

k r = 65,241.27/ 0.00525 = 12,426,909 kip * ft/rad

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232

Appendix F: HHT-a method

In many structural dynamics applications, only low mode response is of
interest. For these cases, the use of implicit unconditionally stable
algorithms is generally preferred over conditionally stable algorithms. For
unconditionally stable algorithms, a time step may be selected
independent of stability considerations, thus results in a substantial saving
of computational effort. In addition to being unconditionally stable, when
only low mode response is of interest, it is often advantageous for an
algorithm to possess some form of numerical dissipation to damp out any
spurious participation of the higher modes. Examples of algorithms
commonly used in structural dynamics which possess these properties are
the Wilson-0 method and the Newmark-(3 method restricted to parameter
1 (V+-)

values of y > - and /? > ^ . The Newmark family of methods allows the

amount of dissipation to be continuously controlled by a parameter other
than the time step. On the other hand, the dissipative properties of this
family of algorithms are considered to be inferior to the Wilson method,
since the lower modes are affected too strongly (Hilber et al. 1977 ).

In the Wilson-0 method, 0 must be selected greater than or equal to 1.37
to maintain unconditional stability. It is recommended to use a value of
9 = 1.42, as further increasing 0 reduces accuracy and further increases
dissipation; but even for 9 = 1.42 the method is highly dissipative. A well
known deficiency of the Wilson-Q method is that it is generally too
dissipative in the lower modes, requiring a time step to be taken that is
smaller than that needed for accuracy. In addition, the Wilson-6 method
tends to damp out the higher modes and could produce large errors when
contributions of higher modes are significant (Wilson 2010 ). Therefore,
the use of Wilson-0 method has limited applications. Despite its
shortcoming, the Wilson-0 method is considered by many to be the best
available unconditionally stable one-step algorithm when numerical
dissipation is desired.

Since it seemed that the commonly used unconditionally stable, dissipative
algorithms of structural dynamics all possessed some drawbacks, in 1977
Hilber, Hughes, and Taylor presented a method called the HHT-a method.
They were looking for an improved one-step method with the following
requirements: a) it should be unconditionally stable when applied to linear
problems, b) it should possess numerical dissipation which can be

ERDC/ITL TR-16-1

233

controlled by a parameter other than the time step, (i.e., no numerical
dissipation should be possible), and c) the numerical dissipation should
not affect the lower modes too strongly. The resulting new algorithm,
which consists of a combination of positive Newmark p-dissipation and
negative a-dissipation, is shown to have improved characteristics when
compared to the Wilson-0 method.

The Hilber, Hughes, and Taylor HHT-a method is a generalization of the
Newmark-P method. The finite-difference equations for the HHT-a
method are identical to those of the Newmark -(3 method. However, the
equations of motion has to be modified using the parameter a, as follows,

mii t + M +1 1 + a)eu t+ M ~ acu > +1 1 + a ) ku t + M ~ aku t = { 1 + a )f t+ ^ (F.l)

where a, (5, and yare free parameters, which govern the stability and
numerical dissipation of the algorithm. If a = o this family of algorithms
reduces to the Newmark family. In this case if y = ^ the algorithms possess

no numerical dissipation whereas if y > - numerical dissipation is present

m 2

and if /? > -—— the new algorithm is unconditionally stable. However, by

appropriately combining negative a-dissipation with particular values of J3
and y, a one-parameter family of algorithms with the attributes previously
enumerated can be developed. Hilber, Hughes, and Taylor in 1977 used /? =

^ tr') ^ 1

—-— and y = - — a. They found by numerical experimentation that the
range of practical interest was — ^ < a < 0. This ensures adequate

dissipation in the higher modes and at the same time guarantees that the
lower modes are not affected too strongly. Finally, if a, P, and y have the

following expressions,- -< a < 0, /? =-, and y = — a, the HHT-a

method is second-order accurate and unconditionally stable. With a = 0 the
HHT-a method reduces to the constant acceleration method. The HHT-a
method is useful in structural dynamics simulations incorporating many
degrees of freedom (DOF), and in which it is desirable to numerically
attenuate the response at high frequencies. Decreasing a below zero
decreases the response at frequencies above provided that /? and y are

defined as above. The procedure of the HHT-a method is summarized in
Table F.l.

ERDC/ITL TR-16-1

Table F.l. HHT-a Method.

1. Initial calculations

1.1 Select At and a, (—^ < a < 0).

1.2 Calculate B = (1 ~ a> and v = -- a.

r 4 1 2

1.3 Solve for it 0 mit 0 = (1 4- a)( p 0 - cit 0 - ku 0 ).

1.4 Calculate c = (1 + a)c ••• k = (1 + a)k

1.5 k = k 4- — c +

/?At /?(At) 2

1.6 a = — m + -c; and b = —m 4- At (— - 1) c.

(3 At (3 ’ 2(3 \2 (3 )

2. Calculations for each time step, /

2.1 c = ac(iii - Wj_i) k = ak{u t - u t _ x ) .*• / = (1 4- a)f i+1 -(14- 2a)/; 4- a/^.

2.2 Apj = / + c + k ■■■ A pi = A pi + aiii + feiij.

2.3 Solve for Ait; from kAu t = Ap;.

2.4 Am i = -^—AUi -\u x 4- At (l - )it;.

1 (3At 1 (3 1 V 2(3 J 1

2.5 Ait/ = —A it/ —— ii, - —it,.

1 (3 {At) 7 - 1 (3 At 1 2(3 1

2.6 u i+1 = Uj + Au f , u i+1 = «i + Aiij, and u i+1 = it; + Ati,.

3. Repetition for the next time step. Replace / by i+1 and implement steps 2.1 to 2.6 for the
next time step.

ERDC/ITL TR-16-1

235

Appendix G: Wilson-6 Method

In Impact_Deck, the solution of the MDOF equations of motion is
obtained by applying the Wilson-0 method. This method, developed by
E.L. Wilson (2002), is a modification of the conditionally stable linear
acceleration method that makes it unconditionally stable. This
modification is based on the assumption that the acceleration varies
linearly over an extended time step S t = 9 At, as shown in Figure G.i. The
accuracy and stability properties of the method depend on the value of the
parameter 0 , which is always greater than 1.

Figure G.I Linear variation of acceleration over normal and
extended time steps.

The numerical procedure can be derived merely by rewriting the basic
relationship of the linear acceleration method. The corresponding matrix
equations that apply to MDOF systems are

Au ; =(At)u ! . + ^Aii 1 . Au i =(At)u i + - ^ il^- J Ail { (G.I)

Replacing At by St and the incremental responses by u t , Silt, and Sii as
shown in Figure G.i gives the corresponding equations for the extended
time step:

ERDC/ITL TR-16-1

236

c. /«. v . , 8f_. « /« \ (St ) 2 .. (St ) 2

8u i = (St)u i +—8u i .-. 8u ; = (8f)u f H - u ; H - 8u ; (G.2)

The right equation in Equation (G.2) can be solved for

8a < = m Su ‘-&- 3ii <

(G.3)

Substituting Equation (G.3) into the left equation of Equation (G.2) gives

Su i = ^ 8 u i- 3 u i~Y U - (G ' 4)

Next, Equations (G.3) and (G.4) are substituted into the incremental (over
the extended time step) equation of motion:

mSii i + cSu i + k t 8u { = 8p t (G.5)

where, based on the assumption that the exciting force vector also varies
linearly over the extended time step.

8p i .=efAp,.J

This substitution leads to

ki 8ui = 8 p {

where:

(OA tf

A Pi

0 A t

m + 3 c

u ,+

0 0 Af
3m H-c

(G.6)

(G.7)

(G.8)

(G.9)

Equation (G.7) is solved for Sui, and Su t is computed from Equation (G.3).
The incremental acceleration over the normal time step is then given by

ERDC/ITL TR-16-1

237

Aii ; . =^8ii, (G.10)

And the incremental velocity and displacement are determined from
Equation (G.i). The procedure is summarized in Table G.i.

Table G.I Wilson’s Method: Nonlinear Systems

1. Initial calculations

1.1 Solve mif'o = p 0 — cu 0 — (/ s ) 0 => u 0 .

1.2 Select At and 9.

1.3 a = -r-m + 3c; and b = 3m + —c.

9At 2

2. Calculations for each time step, /

2.1 Spi = O(Api) + aiii + bill.

2.2 Determine the tangent stiffness matrix k t .

2.3 c + —m.

2.4 Solve for 8u t from k t and 8p t

2 -5 Su t = (eAf)2 Su t flAf u, 3 u t -, and Am, = g Su t .

2.6 Au t = (A i)u t + y A ii t ; and Au t = (A t)u t + ^-ii t + ^-A ii t .

2.7 u i+1 = Ui + Au if u i+1 = iii + Aii if and ii i+1 = u t 4- A ii t .

3. Repetition for the next time step. Replace / by i+1 and implement steps 2.1 to 2.7 for the
next time step.

As mentioned earlier, the value of 0 governs the stability characteristics of
Wilson’s method. If 0 = l, then this method reverts to the linear
acceleration method, which is stable if At < 0.551 T N , where T N is the
shortest natural period of the system. If 0 > 1.37, the Wilson -0 method is
unconditionally stable, making it suitable for direct solution of the
equations of motion and 0 = 1.42 gives optimal accuracy.

ERDC/ITL TR-16-1

238

Appendix H: Member End Release Details for
Load Applied at the End Release Node

Member end release occurs in the Impact_Deck software when there is no
moment transfer from one node to the next, because of an internal pin
from one beam or deck to the next. Member end release details are
discussed in Appendices A, B, and C for the generic case when the load is
not applied at the end release node.

When the load is applied at the end release node, an additional term is
required for the end release equations and the load is altered to
accomplish the lack of moment transfer. This Appendix extends the
equations to include these terms.

For a beam element, the six equilibrium equations in the local reference
system can be written as

1 , 2,...,6

(H.l)

If one end of the member has a hinge, or other type of release that causes
the corresponding force to be equal to zero, Equation (H-i) requires
modification. If we know that a specific value of f n is zero because of a
release, the corresponding displacement u n can be written as:

n—1 h- 12 Jc

u = V-^ U.+ T —

n ^ k J ^ h

j= 1 #v nn j=n+ 1 r Sin

U J+ r n

(H.2)

Therefore, by substitution of equation (H.2) into the other five equilibrium
equations, the unknown u n can be eliminated and the corresponding row
and column set to zero. Or:

f ij = k ij u ij + r V

(H.3)

The terms ^ = r n = 0 and the new stiffness and load terms are equal to:

ERDC/ITL TR-16-1

239

r-r —

1 n k,„

(H.5)

ERDC/ITL TR-16-1

240

Appendix I: Rayleigh Damping

In this Appendix, the implementation of damping in the numerical
procedures used in the computer software Impact_Deck will be discussed.
Rayleigh damping is the method used.

1.1 Method

Consider first mass-proportional damping and stiffness-proportional
damping:

c = a Q m and c=a 1 k (i.i)

Where the constants a 0 and % have units of sec -1 and sec, respectively.
Physically, they represent the damping models shown in Figure I.i for a
multistory building. Stiffness-proportional damping appeals to intuition
because it can be interpreted to model the energy dissipation arising from
story deformations. In contrast, mass-proportional damping is difficult to
justify physically because the mass of the air damping the structure,
compared to the structural mass, is negligibly smaller. Later we shall see
that, by themselves, neither of the two damping models are appropriate
for practical application.

Figure 1.1 (a) Mass-proportional damping; (b) stiffness-proportional damping.

(a) (b)

ERDC/ITL TR-16-1

241

We now relate the modal damping ratios for a system with mass-
proportional damping to the coefficient a 0 . The generalized damping for
the n th mode, Equation (I.i) is

C„ = a 0 M n

( 1 . 2 )

and the modal damping ratio is

(3 =^—

I n o

2 co„

( 1 - 3 )

The damping ratio is inversely proportional to the natural frequency. The
coefficient a 0 can be selected to obtain a specified value of damping ratio in
any one mode, say /?; for the i th mode. Equation (I.3) then gives

a 0 = 2 Pi«i

( 1 . 4 )

When a 0 has been computed, the damping matrix c can be determined
from Equation (I.i). The damping ratio in any other mode, say the n th
mode, is given by Equation ( 1 .3). Similarly, the modal damping ratios for a
system with stiffness-proportional damping can be related to the
coefficient a x . In this case

c„ = a± co 2 n M n and p n

( 1 . 5 )

The damping ratio increases linearly with the natural frequency. The
coefficient a ± can be selected to obtain a specified value of the damping
ratio in any one mode, say /?* for th ej th mode. Equation (I.5) then gives

2 ?,

( 1 . 6 )

When ay has been computed, the damping matrix c can be determined
from Equation (I.i), and the damping ratio in any other mode is given by
Equation (I.5). Neither of the damping matrices defined by the previous
matrices is appropriate for practical analysis of MDOF systems. The
variations of modal damping ratios with the natural frequencies they

ERDC/ITL TR-16-1

242

represent are not consistent with experimental data indicating roughly the
same damping ratios for several vibration modes of a structure.

As a first step toward constructing a classical damping matrix somewhat
consistent with experimental data, we consider Rayleigh damping

c=a 0 m+a 1 k (1.7)

The damping ratio for the n th mode of such a system is

a n 1 a
——+—cd„
2 cd„ 2 "

( 1 . 8 )

The coefficients a 0 and a 1 can be determined from specified damping
ratios and /? ; for the i th and j th modes, respectively. Expressing

Equation ( 1 . 8 ) for these two modes in matrix from leads to

CD,

CD.

a A

( 1 . 9 )

These two algebraic equation can be solved to determine the coefficients
a 0 and a x . If both modes are assumed to have the same damping ratio /?,
which is reasonable based on experimental data, then

„ 2cd,cd, „ 2

a 0 = p- l -^~ a 1 = P-

CD; + CD, CD,. + CDj

( 1 . 10 )

The damping matrix is then computed from Equation ( 1 .7) and the
damping ratio for any other mode, given by Equation ( 1 . 8 ), varies with
natural frequency.

When applying this procedure to a practical problem, the modes i andj
along with specified damping ratios should be chosen to ensure reasonable
values for the damping ratios in all the modes contributing significantly to
the response.

In the Impact_Deck computer program this type of damping model is
implemented to obtain the dynamic response. The user assigns a fixed

ERDC/ITL TR-16-1

243

damping ratio and Impact_Deck estimates the natural frequencies and
calculate the coefficients a 0 and a 1} and assemble the damping matrix of
the system based on Equation (A. 31).

1.2 Impact Deck frequency estimates

For the model of an impact deck or beam directly mounted on a cluster of
piles specified in Appendix A, Impact_Deck estimates the first two natural
frequencies as follows:

co 1

( 1 . 11 )

where:

k = spring transverse stiffness per unit length of the beam = (total
number of springs * linear stiffness of one spring) / total length of
the beam

fh =the mass per unit length of the beam = mass density of the beam *
cross-sectional area of the beam

This estimate is based on SAP2000 models for a similar structure, and
guidance from Chopra (2001).

1.3 Guard Wall frequency estimates

For the model of a guard wall consisting of simply supported beams
specified in Appendix B, Impact_Deck estimates the first two natural
frequencies as follows,

«i =

2 . 5 * k

.\cd 2 =

3.0 *k

( 1 . 12 )

where:

k =the transverse elastic stiffness of the center pile group
m =the total mass of the two beams = mass density of the beam *
cross-sectional area of the beam * total length of the structure

This estimate is based on SAP2000 models for a similar structure.

ERDC/ITL TR-16-1

244

1.4 Flexible Wall frequency estimates

For the flexible wall problem specified in Appendix C, Impact_Deck
estimate the first two natural frequencies as follows:

«i

1 2.0 *m

( 1 . 13 )

where:

k =the transverse elastic stiffness of the center pile group
m =total mass of one beam = mass density of the beam * cross-
sectional area of the beam * length of one beam
m* =total mass of the two beams = mass density of the beam * cross-
sectional area of the beam * total length of the system including
the length of the shear key.

This estimate is based on SAP2000 models for a similar structure.

ERDC/ITL TR-16-1

245

Appendix J: Key Impact.Deck Program
Variables

NSTU mode of operation:

1. Flexible Wall

2. Guard Wall

3. Impact Deck

DS<X,Y,R>() Input displacements (fixed)

FS <X,Y,R> 0 Input force/moment (fixed)

These variables describe a user-defined, piece-wise linear, force
displacement curve for piles.

SPRING_K<X,Y,R>() Computed tangent stiffness of a pile group

SPRING_B<X,Y,R>() Computed force axis intercept a pile group

These variables return the tangent stiffness and force intercept
(deflection=o) at each part of the user-defined, piece-wise linear, force
displacement curve described by DS<X,Y,R>() and FS<X,Y,R>(). By
working from the force intercept, the appropriate force for an absolute
displacement can be computed

SK_SPRING() Current stiffness tangent at pile locations

SB_SPRING() Current stiffness force axis intercept at pile

locations

Returned from the stif¥hess() function, these values return the current
tangent stiffness and intercept that the pile is under. This value takes into
considerations whether the pile is under an unload/reload cycle where the
resisting force is less than the maximum force achieved to that point.

NELASQ Flag for elastic loading condition:

1. Load

ERDC/ITL TR-16-1

246

2. Unload/Reload

uelas<x,y,R>

X_LOAD()

Greatest deflection to this point - for
unload/reload cycle

The x-position of the applied load over time
(along the wall)

This is pre-computed as a linear increment from the start point to the end
point of the simulation, given in block 4 of the input file

U(,)

UDC)

UDD(,)

SK(,)

SM(,)

SKM(,)

SMMQ

SKMMC)

SKMMS(,)

SMMM(,)

The absolute displacement of each pile over
time

The absolute velocity of each pile over time

The absolute acceleration of each pile over time

Element stiffness matrix

Element mass matrix

Global stiffness matrix

Global mass matrix

The global stiffness matrix (sans restraints on
DOF)

Intermediate matrix multi value

The global mass matrix (sans restraints on
DOF)

SCMMC)

ALFAA

BETAA

The global damping matrix
Rayleigh damping term for mass - ao
Rayleigh damping term for stiffness - ai

ERDC/ITL TR-16-1

247

These variables are determined from the natural frequencies of the
structure and □ (EFIl in Block 5 of the Input File)

P_LOAD(,) Force acting at nodes of the element at each

time step (spread to adjacent nodes to the
impact)

FOR_SPRING(,) Element spring forces occurring at each node

at each time step

ELENGTHQ

Length of each wall segment (in the X direction
only)

NRES

Total number of array indexes (~= number of
nodes*number of DOF)

NODF<i, 2,3>() Index for DOF for node number

NOND

Number of nodes

NOTE: may not be the number of pile groups

NPS() Indices for DOF information for pile groups

(Flexible and Guard Walls)

REPORT DOCUMENTATION PAGE

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1. REPORT DATE (DD-MM-YYYY) 2. REPORT TYPE

July 2016

3. DATES COVERED (From - To)

4. TITLE AND SUBTITLE

Simplified Dynamic Structural Time-History Response Analysis of Flexible Approach

Walls Founded on Clustered Pile Groups Using Impact_Deck

5a. CONTRACT NUMBER

5b. GRANT NUMBER

5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)

Barry C. White, Jose Ramon Arroyo, and Robert M. Ebeling

5d. PROJECT NUMBER

5e. TASK NUMBER

5f. WORK UNIT NUMBER

448769

7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES)

Information Technology Laboratory

U.S. Army Engineer Research and Development Center

3909 Halls Ferry Road, Vicksburg, MS 39180-6199;

Department of General Engineering

University of Puerto Rico, Mayaguez, PR 00681

8. PERFORMING ORGANIZATION REPORT
NUMBER

ERDC/ITL TR-16-1

9. SPONSORING / MONITORING AGENCY NAME(S) AND ADDRESS(ES)

U.S. Army Corps of Engineers

441 G. Street NW

Washington, DC 20314-1000

10. SPONSOR/MONITOR’S ACRONYM(S)

11. SPONSOR/MONITOR’S REPORT
NUMBER(S)

12. DISTRIBUTION / AVAILABILITY STATEMENT

Approved for public release; distribution is unlimited.

13. SUPPLEMENTARY NOTES

14. ABSTRACT

Flexible approach walls are being considered for retrofits, replacements, or upgrades to Corps lock structures that have exceeded their
economic lifetime. This report discusses a new engineering software tool to be used in the design or evaluation of flexible approach
walls founded on clustered pile groups and subjected to barge train impact events.

This software tool, Impact_Deck, is used to perform a dynamic, time-domain analysis of three different types of pile-founded flexible
approach walls: an impact deck, an alternative flexible approach wall, and a guard wall. Dynamic loading is performed using impact-
force time histories (Ebeling et al. 2010). This report covers the numerical methods used to create this tool, a discussion of the graphical
user interface for the tool, and an analysis of results for the three wall systems.

The results of analyzing the three wall systems reveals that dynamic evaluations should be performed for these structures because of
inertial effects occurring in the wall superstructure and substructure. These inertial effects can cause overall and individual response
forces that are greater than the peak force from the impact-force time history.

This report also discusses the advantages of load sharing between multiple pile groups in an approach wall substructure. In the case of
Lock and Dam 3, the peak reaction force for any individual pile group was 11% of the peak impact load.

15. SUBJECT TERMS

See reverse

16. SECURITY CLASSIFICATION OF:

17. LIMITATION

18. NUMBER

19a. NAME OF RESPONSIBLE

OF ABSTRACT

OF PAGES

PERSON

a. REPORT

b. ABSTRACT

c. THIS PAGE

19b. TELEPHONE NUMBER (include

Unclassified

264

area code)

Standard Form 298 (Rev. 8-98)

Prescribed by ANSI Std. 239.18

15. SUBJECT TERMS (concluded)

Barge impact
Barge train impact
Flexible lock approach wall
ImpactDeck
ImpactForce
Flexible approach wall
Guide wall
Guard wall
Pile groups
Clustered pile groups
Glancing blow
Impact Deck
Time-history analysis
Force time-history
Dynamic analysis
Dynamic structural analysis
Simply supported beam

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