BRIDGE RAIL DESIGN PROCEDURE by EMAD BADIEE · PDF file · 2015-02-18BRIDGE RAIL...

BRIDGE RAIL DESIGN PROCEDURE

by

EMAD BADIEE

NASIM UDDIN, CHAIR IAN EDWARD HOSCH

LEE MORADI

A THESIS

Submitted to the graduate faculty of The University of Alabama at Birmingham,

in partial fulfillment of the requirements for the degree of Master of Science

BIRMINGHAM, ALABAMA

2014

Copyright by Emad Badiee

2014

iii

BRIDGE RAIL DESIGN PROCEDURE

EMAD BADIEE

CIVIL ENGINEERING

ABSTRACT

The AASHTO Bridge Specifications recommend a yield line theory analysis to

determine the structural capacity of concrete bridge railing based on static strength of

concrete. However, this analysis technique has been shown to significantly underestimate

the capacity of concrete bridge rails to withstand high speed truck impacts. Traditionally

this shortcoming has been mitigated by artificial reductions in bridge rail design loads

implemented into the design specifications. Fear of litigation associated with failure of a

bridge rail to contain and redirect an errant vehicle has made continuing this policy

unacceptable for most state highway agencies. On the other hand, existing barrier design

guidelines contained in the Bridge Specifications are based upon National Cooperative

Highway Research Program (NCHRP) Report 350. This document has been superseded

by the Manual for Assessing Safety Hardware (MASH). The updated performance

guidelines incorporate heavier vehicles, higher impact angles, and in one case, higher

impact speeds. Full-scale crash testing has shown that the new testing criteria will require

stronger and taller barriers. One study has attempted to generate new height and design

load requirements for inclusion in the updated Bridge Design Specifications. The load

recommended for implementation proved to be extremely high and was not well received

by AASHTO’s T7 committee on Guardrails and Bridge Rails. Thus, there is a national

need for a more thorough evaluation of bridge rail design loads and minimum barrier

heights required to meet the MASH guidelines. This thesis presents an improved method

iv

based on modified Yield Line Theory and dynamic strength of concrete to estimate the

design impact loads to realistic levels without adjustment of the underlying analysis

technique. The net effect of applying the new method to the design would be large

decreases in the size and cost of bridge railing necessary to withstand the elevated loads.

The objective of the research proposed herein includes: (1) Developing improved

methods for estimating the structural capacity of bridge rails and cantilevered deck

systems based on dynamic strength of concrete, and including the contribution of

deflection of deck overhang, moment of inertia of the barrier and deck overhang sections,

and mass of the vehicle and barrier and (2) Identifying appropriate design loads for use

in the new methods that are representative of MASH recommended crash test conditions

TL-2 through TL-5.

Keywords: NCHRP, MASH, AASHTO, Bridge railing, Yield Line Theory, Moment of Inertia.

v

ACKNOWLEDGMENTS

I am greatly indebted to my advisor and committee chair, Dr. Nasim Uddin, for

all of his time, support, advice, and relentless patience in the development of this project.

It has certainly been a rewarding experience, and I am grateful for the opportunity

provided to me.

As well I wish to thank Dr. Dean L Sicking and Dr. Lee Moradi for their

cooperation, expertise, and insight. I also wish to thank Dr. Ian Hosch for all of his time

and guidance.

Last but not least, I express my gratitude to my family, especially my parents and

brothers, for all of their love and support.

Finally, I thank UAB for this opportunity, which allowed me to grow personally

and professionally.

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TABLE OF CONTENTS

Page

ABSTRACT...................................................................................................................... iii

ACKNOWLEDGMENTS……………………………………………………………….. v

1 INTRODUCTION .................................................................................................. 1

1.1 Background ............................................................................................................. 1

1.2 Problem Statement .................................................................................................. 7

1.3 Objective ................................................................................................................. 8

2 LITERATURE REVIEW ..................................................................................... 10

2.1 Guardrail Design ................................................................................................... 10

2.2 Barrier Strength ..................................................................................................... 14

3 WORK METHOD ................................................................................................ 15

3.1 Objective ............................................................................................................... 15

3.2 Test Levels ............................................................................................................ 16

3.3 Modified Yield Line Method ................................................................................ 17

3.3.1 Unit Mass Velocity ....................................................................................... 17

3.3.2 Distributed Impact Force .............................................................................. 18

3.3.3 Dynamic Increase Factor .............................................................................. 20

3.3.4 Moment of Inertia ......................................................................................... 23

3.3.4.1 Barrier .........................................................................................23

3.3.4.1.1 Segment I......................................................................................... 24

3.3.4.1.2 Segment II ....................................................................................... 27

3.3.4.1.3 Segment III ...................................................................................... 30

3.3.4.1.4 Barrier Section Moment of Inertia .................................................. 33

3.3.4.2 Deck Overhang ...........................................................................34

vii

TABLE OF CONTENTS (Cont.)

Page

3.3.5 Displacement................................................................................................. 37

3.3.5.1 Barrier .........................................................................................37

3.3.5.2 Deck Overhang ...........................................................................38

3.3.5.3 Superposition ..............................................................................41

3.3.6 Strain Energy Absorption due to Total Horizontal Displacement ................ 41

3.3.7 Moment Capacity of the Barrier ................................................................... 42

3.3.7.1 Vertical Moment Capacity, 𝑀𝑤 .................................................42

3.3.7.1.1 Segment I......................................................................................... 42

3.3.7.1.2 Segment II ....................................................................................... 43

3.3.7.1.3 Segment III ...................................................................................... 44

3.3.7.2 Horizontal moment capacity, 𝑀𝑐 ................................................45

3.3.7.2.1 Segment I......................................................................................... 45

3.3.7.2.2 Segment II and III ........................................................................... 46

3.3.7.3 Top Beam Moment Capacity ......................................................46

3.3.8 Internal Virtual Work along Yield Line, Eyield ........................................... 46

3.3.9 External Virtual work by Applied Load ....................................................... 48

3.3.10 Critical length of Yield-Line failure pattern, Lc ......................................... 49

3.3.11 Nominal Railing Resistance to Transverse Loads, Rw ............................... 52

4 ENERGY METHOD ............................................................................................ 57

4.1 Objective ............................................................................................................... 57

4.2 Moving Vehicle Energy, SI .................................................................................. 58

4.3 Barrier Strain Energy ............................................................................................ 59

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Page

4.3.1 Barrier Strain Energy Capacity in Elastic Region, E1 .................................. 59

4.3.2 Absorbed Energy by the Barrier, Δ1 ............................................................. 60

4.4 Absorbed Energy by Barrier in Plastic Region, Δ2 .............................................. 60

4.5 Absorbed Energy by Vehicle Deformation, Δ3 .................................................... 62

4.6 Deck Overhang Strain Energy .............................................................................. 66

4.6.1 Deck Overhang Strain Energy Capacity in Elastic Region, E4 .................... 66

4.6.2 Absorbed Energy by Deck Overhang, Δ4 ..................................................... 67

5 LS-DYNA SIMULATION ................................................................................... 69

5.1 Implementation of LS-DYNA .............................................................................. 69

5.2 NCAC Model ........................................................................................................ 69

5.2.1 NCAC Single Unit Truck.............................................................................. 70

5.2.2 NCAC Rigid Barrier ..................................................................................... 72

5.2.3 Objective of NCAC Model ........................................................................... 74

5.3 Proposed Model .................................................................................................... 77

5.3.1 Deck Overhang ............................................................................................. 80

5.3.2 NEW JERSEY Concrete Barrier .................................................................. 80

6 RESULTS AND DISCUSSION ........................................................................... 88

6.1 Work Method ........................................................................................................ 88

6.1.1 Results Compression ..................................................................................... 91

6.2 Energy Method...................................................................................................... 93

6.2.1 Moving Vehicle Energy, IS .......................................................................... 93

6.2.2 Barrier Strain Energy .................................................................................... 93

6.2.2.1 Barrier Strain Energy Capacity, 𝐸1 ............................................93

6.2.2.2 Absorbed Energy by Barrier in Elastic Region, 𝛥1 ....................94

6.2.2.3 Absorbed Energy by Barrier in Plastic Region, 𝛥2 ....................95

6.2.3 Absorbed Energy by Vehicle, Δ3................................................................. 95

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Page

6.2.4 Deck Overhang Strain Energy ...................................................................... 97

6.2.4.1 Deck Overhang Strain Energy Capacity, 𝐸4 ..............................97

6.2.4.2 Absorbed Energy by Deck Overhang in Elastic Region, 𝛥4 ......99

6.3 Results from Proposed Model ............................................................................... 99

7 SUMMARY AND CONCLSION ...................................................................... 109

7.1 Work Method ...................................................................................................... 109

7.1.1 Work Method Conclusions ......................................................................... 110

7.2 Energy Method.................................................................................................... 111

7.2.1 Energy Method Conclusions ....................................................................... 112

7.3 LS-DYNA Model................................................................................................ 113

7.3.1 LS-DYNA Model Conclusions ................................................................... 113

7.4 Recommendations for future studies ................................................................. 114

LIST OF REFERENCES ................................................................................................ 116

APPENDIX A ................................................................................................................. 117

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LIST OF TABLES

Page

Table 1: Test Levels Configurations. (AASHTO, A13.7.2-1.) ........................................... 2

Table 2: AASHTO Specifications Test levels Details. (AASHTO, A13.7.2-1.) .............. 53

Table 3: AASHTO Specifications Test Levels Details..................................................... 90

Table 4: Work Method Results Comparison .................................................................... 92

Table 5: Work Method Results Comparison .................................................................. 111

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LIST OF FIGURES

Page

Figure 1: Test Level 2 Vehicle Impact. (After Sheikh, 2007) ............................................ 3

Figure 2: Test Level 3 Vehicle Impact. (After Bligh, 2010) .............................................. 4

Figure 3: Test Level 4 Vehicle Impact. (After Sheikh, 2011) ............................................ 5

Figure 4: Concrete NEW JERSEY Type Barrier. (After Barker, 2013) ............................. 6

Figure 5: Front View of Yield Lines Failure Pattern. (After Hirsh, 1978) ......................... 7

Figure 6: Yield Line Pattern due to Collision Force. (After Hirsch, 1978.) ..................... 11

Figure 7: External Virtual Work Done by Vehicle Collision. (After Calloway, 1993) .... 11

Figure 8: Top View of Plastic Hinge for Top Beam. (After Calloway, 1993) ................. 12

Figure 9: Moment Capacities of Barrier Wall. (After Calloway, 1993) ........................... 13

Figure 10: Top View prior to the Impact .......................................................................... 18

Figure 11: Impact Moment Top View .............................................................................. 19

Figure 12: Strain Rate According to Real Loads.(After Pajak, 2011) .............................. 21

Figure 13: Strain Rate Effect on Compressive Strength of Concrete.(After Pajak, 2011.)

........................................................................................................................................... 21

Figure 14: Strain Rate Effect on Tensile Strength of Concrete.(After Pajak, 2011.) ....... 22

Figure 15: 3D View of Impact Force on Barrier ............................................................... 23

Figure 16: Barrier Segments ............................................................................................. 24

Figure 17: Barrier Top Segment Vertical Rebar in XY Plane .......................................... 25

Figure 18: Plan View of Barrier Top Segment in ZX Plane ............................................. 26

Figure 19: Vertical Rebar of Barrier Second Segment in XY Plane ................................ 28

Figure 20: Plan View of Barrier Middle Segment in ZX Plane ........................................ 29

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LIST OF FIGURES (Cont.)

Page

Figure 21: Barrier Bottom Segment with Vertical Rebar in XY plane............................. 31

Figure 22: Plan View of Barrier Bottom Segment in ZX Plane ....................................... 32

Figure 23: SUPER POSITION Method for Impact Force ................................................ 34

Figure 24: 3D View of Barrier and Deck.......................................................................... 35

Figure 25: Deck Overhang Section in ZY Plane .............................................................. 35

Figure 26: Barrier Elastic Deflection due to Vehicle Collision Force .............................. 38

Figure 27: Deck Overhang Vertical Deflection ................................................................ 39

Figure 28: Deck Overhang Deflection Configuration ...................................................... 40

Figure 29: Superposition Method for Barrier in Horizontal Deflection ........................... 41

Figure 30: Barrier Segment I in XY Plane. (After Barker, 2013) ..................................... 43

Figure 31: Barrier Segment II in XY Plane. (After Barker, 2013) ................................... 43

Figure 32: Barrier Segment III in XY Plane. (After Barker, 2013) .................................. 44

Figure 33: Front View of Yield Line Failure Pattern. (After Calloway, 1993) ................ 45

Figure 34: Top View of Yield Line Failure Pattern. (After Calloway, 1993) .................. 47

Figure 35: External Virtual Work by the Impact Load. (After Calloway, 1993) ............. 48

Figure 36: Yield Line Pattern Front View. (After Barker, 2013) ..................................... 52

Figure 37: Transferred Collision Force Between Barrier and Deck. (After Barker, 2013)

........................................................................................................................................... 54

Figure 38: Moving Vehicle Kinetic Energy ...................................................................... 59

Figure 39: Stress-Strain Curve of Concrete ...................................................................... 60

Figure 40: External Virtual Work by the Impact Load. (After Calloway, 1993) ............. 61

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Page

Figure 41: Initial Vehicle Deformation............................................................................. 63

Figure 42: Final Vehicle Deformation .............................................................................. 64

Figure 43: Vehicle Deformation Side View ..................................................................... 65

Figure 44: Vehicle Deformation of TL-4. (After Sheikh, 2011) ...................................... 66

Figure 45: Single Unit Truck Top View ........................................................................... 70

Figure 46: Single Unit Truck Side View .......................................................................... 71

Figure 47: Single Unit Truck Front View ......................................................................... 72

Figure 48: NCAC Barrier Top View ................................................................................ 73

Figure 49: NCAC Barrier Front View .............................................................................. 73

Figure 50: NCAC Barrier 3D View .................................................................................. 74

Figure 51: NCAC Model Top View ................................................................................. 75

Figure 52: NCAC Model 3D View ................................................................................... 76

Figure 53: NCAC Model Side View ................................................................................. 77

Figure 54: Stress-Strain Curve of 60 ksi Steel .................................................................. 79

Figure 55: Section Details. (After Barker, 2013) .............................................................. 81

Figure 56: Model Section.................................................................................................. 82

Figure 57: Model 3D Section ............................................................................................ 83

Figure 58: Section Element Formation ............................................................................. 84

Figure 59: Model 3D view ................................................................................................ 85

Figure 60: Model Top View ............................................................................................. 85

Figure 61: Model Side View ............................................................................................. 86

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Page

Figure 62: Equivalent Distributed Force........................................................................... 87

Figure 63: Change in Kinetic Energy of the Vehicle Based on TL-4 ............................... 97

Figure 64: 1 Foot Strip of Deck Overhang Section in ZY Plane ...................................... 98

Figure 65: Maximum Displacement of Barrier ............................................................... 100

Figure 66: Barrier Maximum Displacement with Scale of 50 ........................................ 100

Figure 67: Maximum Displacement of Deck Overhang ................................................. 101

Figure 68: Deck Overhang Maximum Displacement with Scale of 50 .......................... 102

Figure 69: Maximum Effective Stress in Barrier ............................................................ 103

Figure 70: 3D View of Barrier Maximum Effective Stress ............................................ 104

Figure 71: Maximum Effective Stress in Deck Overhang .............................................. 105

Figure 72: Bottom View of Deck Overhang, Maximum Effective Stress ...................... 106

Figure 73: Axial Force Resultant Distribution ................................................................ 107

Figure 74: Rebar Front View .......................................................................................... 108

1

1 INTRODUCTION

1.1 Background

Recently, the application of using median and parapet to divide the highways and

roadways has become an important subject in highway designs. The main object of

median is to provide a recovery area for errant vehicles to reach steady-state without

interrupting traffic flow. However, many structures have been installed in the median, for

instance, bridge supports and piers. Although these types of structures are often located a

short distance from the roadway, they can cause serious accidents in the median as well

as along the roadway. Barriers, a reasonable distance from these structures are installed in

order to minimize hazards and avoid serious accidents. As an example, steel guardrail

envelopes are used to protect vehicles from impacting the bridge structure, mostly on

piers faces, and upstream. In order to protect the structures from vehicle collision, the

design of all barriers and guardrails must be such that the vehicle would not be able to

penetrate and pass through the barrier and reach the structure surface. In the case of semi-

rigid barriers, full scale crash testing, based on the AASHTO manual, determines the

placement of the barrier (Reid, 2008). There are six test levels defined in AASHTO

Standard Bridge Specifications that are vary in type of vehicle, impact angle, and vehicle

velocity. Each test level presents a certain vehicle crashing the barrier with certain impact

angles and velocities.[A13.7.3.1].

2

Table 1: Test Levels Configurations. (AASHTO, A13.7.2-1.)

In the table above, W represents the vehicle mass in kips; B is the location of the

center of gravity from the front edge of vehicle; θ represents the crash angle in degrees;

and TL-1 through 5 represent the test levels. Test level 1 through 3 does not apply to

tractor-trailers and trailers. Figure 1 and Figure 2 represent test level 2 and 3 procedures

respectively for 4.5 kips pickup trucks crashing the barrier with a velocity of 45 mph and

with 25 degrees as the angle of crash. Figure 3 presents AASHTO test level 4.

3

Figure 1: Test Level 2 Vehicle Impact. (After Sheikh, 2007)

4

Figure 2: Test Level 3 Vehicle Impact. (After Bligh, 2010)

5

Figure 3: Test Level 4 Vehicle Impact. (After Sheikh, 2011)

Overtime, various types of barriers and parapets have been produced with

specified missions and purposes. The main objective of concrete barriers in vehicle

6

collision is to redirect and to control the vehicle such that said vehicle would not be able

to redirect the flow of traffic. Therefore, the barrier must meet the design criteria

necessary to absorb the impact energy during vehicle collision as well as to redirect the

vehicle in a controlled manner. For this purpose the barrier must satisfy both geometric as

well as strength design. Geometric design is based on the redirection of the vehicle

whether it is in a controlled manner or not, whereas strength design depends on the

vehicle type and its velocity and its computation is based on traffic flow and test level in

formation.

NEW JERSEY type concrete barriers are a type of concrete barrier that meets the

criteria of TL 4. Figure 4 represents the appropriate dimensions of NEW JERSEY barrier

based on TL 4.

Figure 4: Concrete NEW JERSEY Type Barrier. (After Barker, 2013)

The strength design of the barrier is based on the yield line equation and the limit

states. It has been assumed that the vehicle collision produces distributed impact force 𝐹𝑡,

7

along the length 𝐿𝑡, which causes the yield lines failure pattern in the barrier (Barker,

2013).

Figure 5: Front View of Yield Lines Failure Pattern. (After Hirsh, 1978)

1.2 Problem Statement

The yield line method indicates that the external virtual work done by the applied

loads is equal to the internal virtual work done by the resisting moments along the yield

lines. The calculated impact force 𝐹𝑡, based on the yield line equation, is either equal to

or greater than the real impact force of the truck. Also, in case of barriers installed on a

bridge deck overhang, since the deck overhang strain energy absorption is neglected, the

yield line design does not reflect the actual condition of the impact. In reality large

amounts of the vehicle impact energy will be absorbed by the deck overhang vertical

deflection. The yield line design calculates the required capacity of the barrier at a higher

value than what it should be; therefore, the yield line design can be considered as more

conservative.

8

1.3 Objective

The objective of this research is to first propose a modified yield line method

based on work theory by taking into account the effect of deck overhang vertical

deflection in yield line method, and to calculate the revised critical length of the barrier in

order to calculate the capacity of said barrier.

The second objective of this research is to propose an energy based method to

determine the contribution of each component in the system in terms of their energy

absorption and to compare that to the energy absorption capacity of each component by

using the conservation of energy equation for impact due to transferred energy and all

losses.

The third objective of this research is to develop a LS-DYNA model in order to

validate proposed methods, also the model developed can be used as a simulation tool for

other aspects of test levels and the interaction of barrier and deck overhang systems in

future research.

In order to do that, a modified version of yield line analysis with contribution

from deck overhang and barrier deflections in the elastic region of section is presented in

chapter 3. Since TL-4 is a type of dynamic loading, this procedure is followed by the

energy method presented in chapter 4, a method based on conservation of energy.

Chapter 5 indicates the simulation of TL-4 in LS-DYNA code with equivalent force

applied to the model. In conclusion, the results of all 3 methods are summarized and

compared in chapter 7.

9

This is the first time that a research has been done on NEW JERSEY concrete

barriers’ capacity, that research proved that the current design is more conservative than

what it needs to be. Also for the first time, LS-DYNA model of AASHTO Standard

NEW JERSEY barrier has been developed in order to validate the analytical methods to

be applied to future research on barriers and cantilever overhang systems as well as

different aspects of test levels.

10

2 LITERATURE REVIEW

2.1 Guardrail Design

Over the years, various types of guardrails with different purposes have been

developed. The main purpose of concrete barriers is to absorb collision impact energy

and to redirect the vehicle in a controlled manner. Guardrail design consists of geometric

design as well as strength design. Geometric design refers to its aesthetic (a branch of

philosophy) and also governs the redirecting of the vehicle after the collision whether it is

in a controlled manner or not.

The strength design of the barriers depends on the size, geometry and velocity of

the vehicle and the traffic volume of either the bridge or roadway. For a given condition

of the roadway, the barrier strength and performance can be selected from the AASHTO

Standard Bridge Specifications [A13.7.2].

Hirsh (1978) analyzed the lateral load capacity for the barriers with uniform

thickness. He expressed the strength of the barrier to lateral load based of the formation

of yield line analysis and limit states. Yield line implies an assumed failure pattern of the

barrier caused by the vehicle impact force 𝐹𝑡, over the length of distributed impact

force 𝐿𝑡. For an assumed yield line failure pattern, the external virtual work done by the

applied load - which is the vehicles impact force - must be equate to the internal virtual

work done by the resisting moments of the barrier along the yield line. The calculated

11

applied load based on the yield line method is either equal to or greater than the actual

impact force. Therefore, it is important to minimize the impact load (Hirsch, 1978).

Figure 6: Yield Line Pattern due to Collision Force. (After Hirsch, 1978.)

Calloway (1993) presented the equation for external virtual work done by the applied

load with respect to vehicle horizontal collision force 𝐹𝑡, deformation of the barrier in

horizontal direction δ, critical length of the barrier 𝐿𝑐 and the length of distributed

collision force 𝐿𝑡. Callaway implied that the shaded area of Figure 7 presents the integral

of total horizontal deformation through the distributed length of vehicle collision.

Figure 7: External Virtual Work Done by Vehicle Collision. (After Calloway, 1993)

12

The internal virtual work along the yield lines is the summation of all rotations

and moments caused by the barrier displacement. It has been assumed that the barrier acts

as a rigid wall so that all the rotations occur in yield line paths. The rotation of the top

part of the barrier can be expressed as,

Θ = tan Θ= 2δ𝐿𝑐

(2-1)

The barrier can be divided into 2 segment, top beam and the uniform thickness wall

below that. The top beam of the barrier develops plastic moment of 𝑀𝑏

Figure 8: Top View of Plastic Hinge for Top Beam. (After Calloway, 1993)

The horizontal reinforcement of the wall develops moment resistance in vertical

direction 𝑀𝑤, and the vertical reinforcement of the wall develops moment resistance in

horizontal direction 𝑀𝑐. These two moment capacities develop inclined moment

capacity 𝑀𝛼, along the yield line.

13

Figure 9: Moment Capacities of Barrier Wall. (After Calloway, 1993)

Therefore, the nominal railing resistance is achieved by adding the virtual works

provided by moments.

The nominal railing resistance to transverse impact collision force can be

expressed by equating the external virtual work by applied loads to internal virtual work

along the yield lines. In order to minimize the calculated vehicle collision, the yield line

equation must be written based on vehicle collision force and be differentiated with

respect to critical length. The next step is to set the result equal to zero to find the critical

length of the barrier. The vehicle collision force can be achieved by substituting the

critical length in the yield line equation. The achieved value of the vehicle collision force

is denoted as nominal railing resistance of the barrier to transverse loads 𝑅𝑤 (Calloway,

1993).

14

2.2 Barrier Strength

Calloway (1993) investigated barriers strength with non-constant thickness based

on yield line approach. The equation for railing resistance was developed by integrating

the various moments and rotations along the thickness. Barker (2013) compared

Calloway’s and Hirsch’s approaches in terms of critical length and nominal railing

resistance to transverse loads, in order to obtain barrier wall moment capacities. The

recommended procedure based on the comparison was to use Hirsch’s equations with the

average value for vertical and horizontal moment capacities. In case of using the average

values in Hirsch’s equation, Calloway concluded that the calculated nominal railing

resistance to transverse loads is 4% less than the actual nominal railing resistance, which

leads to a conservative design (Barker, 2013).

15

3 WORK METHOD

3.1 Objective

Yield Line approach is an energy method used in AASHTO Standard Bridge

Specifications (American Association of State Highway and Transportation Officials), in

order to design and to calculate the moment capacity of the barrier. The solution of this

approach can be obtained by equating the external virtual work done by the applied loads

to the internal virtual work done by resisting moments along the yield lines.

W = W yield (3-1)

In the equation above, W represents the external virtual work done by applied load and

W yield is the internal virtual work done by resisting moments.

However, in the case of barriers settled on the bridge deck overhangs,

conservative design of AASHTO Standard Bridge Specifications does not reflect the

strain energy absorption of the deck overhang in a vertical direction due to the vehicle

impact in the Yield Line approach, so that the deck overhang energy absorption during

vehicle impact has been completely ignored. Based on AASHTO design criteria, a new

modification can be applied to the Yield Line approach due to the strain energy absorbed

in the deck overhang during vehicle impact. The modification can be developed by

adding a new term “Strain Energy Absorption due to the deck overhang deflection” to the

Yield Line approach. Therefore, the modified equation can be expressed as

W − Wd = W yield (3-2)

16

In equation above Wd presents the Strain Energy Absorption due to the deck overhang

deflection.

3.2 Test Levels

Six different test levels are defined in AASHTO Standard Bridge Specifications

based on different vehicles, angles and velocities for designing barriers. AASHTO design

for NEW JERSEY concrete barriers is based on Test Level four (TL-4) which represents

an 18 kips single unit truck hitting the barrier with the velocity of 50 miles per hour and

with an angle of impact of 15 degrees.

Table 2: Test Levels Configurations. (AASHTO, A13.7.2-1.)

17

3.3 Modified Yield Line Method

3.3.1 Unit Mass Velocity

Since vehicle impact is a type of dynamic loading applied to barriers, it is possible

to use conservation of momentum and energy based equation, in order to obtain strain

energy absorption due to the deck overhang deflection. Momentum is the quantity of

motion of a moving body, measured as a product of its mass and velocity. So that the

conservation of momentum equation for TL-4 can be written as:

m1v1 sin Θ + m2v2= (m1 + m2) V (3-3)

𝑚1 represents the mass of the vehicle 𝑣1 is the initial velocity of the vehicle, 𝛩 is the

angle of the impact 𝑚2 is the unit mass of the barrier, 𝑣2 is the initial velocity of the

barrier before impact – which is zero – and V is the combined mass velocity

perpendicular to the barrier, where the barrier and vehicle can be considered as a unit

mass for a short period of time after the impact.

In order to calculate the Combined Mass Velocity, the left side of the

conservation of momentum equation, which is the total momentum prior to the impact

must be divided by the total mass.

V=m1v1 sin Θm1+m1

(3-4)

18

Figure 10: Top View prior to the Impact

3.3.2 Distributed Impact Force

Impulse is change in momentum over time and can be expressed for both barrier

and vehicles as well. Vehicle impulse is:

m1v1 + ∫ Ftdt = m1V (3-5)

And for the barrier is:

m2v2 + ∫ Ftdt = m2V (3-6)

Where Ft is the vehicle impact force perpendicular to the barrier, and dt the initial impact

duration is equal to Δt, the time that the barrier and vehicle can be considered as a unit

mass.

dt = Δt (3-7)

19

So that by rewriting equation (3-6),

m2v2 + FtΔt = m2V (3-8)

In order to calculate the impact force, 𝐹𝑡, barrier energy before impact must go to the

right side of the equation,

FtΔt = m2V − m2v2 (3-9)

Next divide the equation by the impact time, Δt.

Ft=m2V− m2v2

Δt (3-10)

Since the barrier does not have any motion prior to impact, the term m2v2 is equal to zero

and the equation can be rewritten as:

Ft=m2V

Δt (3-11)

The Combined Mass Velocity V, has already been calculated.

Figure 11: Impact Moment Top View

20

3.3.3 Dynamic Increase Factor

Pajak (2011) proposed that concrete in compression, for all range of strain rates

must be investigated in two domains of strain rate. The first domain is the strain rate that

the answer is changing which is called the transition strain rate. In this region, the

dynamic increase factor, DIF, can achieve up to 1.8 – DIF is defined as the ratio of

dynamic strength to quasi static strength –. The second domain is denoted as pronounced

strength where DIF in this domain is equal to 3.5. There is a shift in DIF for higher strain

rates ( 10 1/s).

The sensitivity of concrete in tension is significantly different than in

compression. The DIF factor in tension can reach 13. Generally concrete behavior under

different tensile strain rates is more uniform than in compression. In overall view, the

author implied that the behavior of concrete in tension and compression in smaller strain

rates (up to 10−1) can be considered the same and that the significant difference starts at

higher strain rates. The author also proposed that the size and geometry of concrete

specimens probably do not have any effect on strain rate. The values of DIF factors are

presented based on normal strength concrete.

In order to reflect the dynamic loading in the analysis for concrete, modulus of

elasticity and compressive strength of the concrete must be multiplied by the DIF factor.

The first step is to determine the strain rate based on the loading type. The author

proposed a chart to determine the strain rate in dynamic loading based on loading type.

As shown in Figure 12 the chart represents that the strain rate, according to real loads, is

approximately 10−4 to 10−3 for vehicle impact (Pajak, 2011).

21

Figure 12: Strain Rate According to Real Loads.(After Pajak, 2011)

Strain rate has effect on both compressive and tensile strength of concrete. Figure 13 and

Figure 14 present this effect on compressive and tensile strength of concrete respectively.

Figure 13: Strain Rate Effect on Compressive Strength of Concrete.(After Pajak, 2011.)

22

Figure 14: Strain Rate Effect on Tensile Strength of Concrete.(After Pajak, 2011.)

Change in strain rate curves for both tensile and compressive strength of concrete,

is approximately equal up to 10−1 [1/s] so that the DIF is approximately 1.05 to 1.20 and

in this study, it has been conservatively assumed that the DIF is equal to 1.05 (Pajak,

2013).

Average DIF=1.05 �Enew = 1.05 E

fc new/ = 1.05 fc

/ (3-12)

23

3.3.4 Moment of Inertia

3.3.4.1 Barrier

Since the impact force, 𝐹𝑡, is perpendicular to the barrier face, the barrier moment

of inertia must be calculated about the rotation axis which is the Z axis in the XZ plane.

Also the barrier thickness varies from top to bottom (Y axis), so that, in order to calculate

the moment of inertia, the barrier must be divided into three segments with three

individual moments of inertia and must use the weighted-mean for the total moment of

inertia for the section. Figure 16 represents different segments of a barrier.

Figure 15: 3D View of Impact Force on Barrier

24

Figure 16: Barrier Segments

3.3.4.1.1 Segment I

Figure 17 represents the top segment as a trapezoidal consists of two heights, 6

inches at the top and 8 inches at the bottom. In order to calculate the moment of inertia in

ZX plane, the segment must be simplified to a rectangular segment with height of the

average of the trapezoid’s two heights. Also the only rebar contributing to strain energy

absorption about the Z axis are vertical rebar and stirrups.

Modified height = 6+82

= 7 in.

25

Figure 17: Barrier Top Segment Vertical Rebar in XY Plane

Since the strain energy will be calculated for a unit length of the barrier, the

length of the section is equal to 1 foot (12 inches), so that, based on Standard NEW

JERSEY type barrier; the modified segment will be a rectangle with 7 inches in height

and 12 inches in width. Figure 18 represents the modified section.

26

Figure 18: Plan View of Barrier Top Segment in ZX Plane

The distance between the bottom edge of the section to the center of gravity of the

bottom rebar, based on 2 inches cover, is 2.25 inches and the distance from the bottom

edge to the center of gravity of the top rebar is approximately 4.75 inches; also the area of

No.4 rebar is 0.20 in2. The total amount of steel area used in this section is

(2 × 0.20) + (4 × 0.202

) = 0.8 in2

In order to calculate the center of gravity, the steel area must be transformed to

concrete. Therefore, n, which is the ratio between the modulus of elasticity of steel to the

modulus of elasticity of concrete, must be multiplied by the steel area. Since rebar is 60

ksi steel, n in expressed as,

n= EsteelEconcrete

= 29000 ksi30450 ksi

= 8.044

The center of gravity of the section is,

CG =∑𝐴𝑑∑𝐴

(3-13)

27

A represents the area of rebar and concrete respectively and d represents the distance

between the center of gravity and the bottom edge of the section of rebar. Therefore,

CG =( A d)rbr.+( A d)conc.A rbr.+Aconc.

(3-14)

So that,

CG =(2 × 8.044 × 0.20 × 4.75)+(2 ×8.044 × 0.20 × 2.25)+(7 × 12 ×3.5)(2 × 8.044 × 0.20)+(2 × 8.044 × 0.20)+(7 ×12)

= 3.5 in.

Moment of inertia of the section is equal to,

I = 𝑏 ℎ3

12 + (A d2)𝑐𝑜𝑛𝑐. + (A d2)𝑟𝑏𝑟. (3-15)

The distance from the center of gravity of the concrete to the section center of gravity is

zero, so that the second term of the above equation is equal to zero.

I= 12 × 73

12 + [2 × 0.20 ×8.044 × (3.5 − 2.25)2] + [2 × 0.20 ×8.044 × (4.75 − 3.5)2] =

353.055 in4

ft�

In order to convert the moment of inertia to ft4

ft� , it must be multiplied by 0.083334. So,

I = 353.055 in4

ft� = 0.017026 ft4

ft�

3.3.4.1.2 Segment II

28

The figure below represents the mid segment of a trapezoid with heights of 8

inches at the top and 15 inches at the bottom. In order to calculate the moment of inertia

in the ZX plane, the segment must be simplified to a rectangular segment with its height

the average of the trapezoid’s two heights.

Figure 19: Vertical Rebar of Barrier Second Segment in XY Plane

The only rebar contributing in strain energy absorption about the Z axis are

vertical rebar and stirrups in the Y direction; also, the effect of stirrups are negligible

since that effect is small.

Modified height = 6+82

= 7 in.

As was the case with the top section, the strain energy will be calculated for a unit

length of barrier. The length of the section will be equal to one foot (12 inches), so that

based on AASHTO Standard NEW JERSEY type barrier, the modified segment will be a

rectangle with 11.5 inches in height and 12 inches in wide. Figure 20 presents the

modified section.

29

Figure 20: Plan View of Barrier Middle Segment in ZX Plane



edge to the center of gravity of the top rebar is approximately 9.25 inches, and the area of

No.4 rebar is 0.20 in2. The total amount of steel area used in this section is,

(2 × 0.20) + (4 × 0.202

) = 0.8 in2


CG =∑Ad∑A

(3-16)


between the center of gravity and the bottom edge of the section for rebar. Therefore;


(3-17)

30

So that,

CG =(2 × 8.044 ×0.20 × 9.25)+(2 ×8.044 × 0.20 × 2.25)+(7 × 12 ×5.75)(2 ×8.044 × 0.20)+(2 ×8.044 × 0.20)+(11.5 ×12)

= 5.75 in.

The moment of inertia of the section is,

I = 𝑏 ℎ3

12 + (A d2)𝑐𝑜𝑛𝑐. + (A d2)𝑟𝑏𝑟. (3-18)

The distance from the center of gravity of concrete to the section center of gravity is zero,

so, the second term of the equation above is zero.

I= 12 × 11.53

12 + [2 × 0.20 ×8.044 × (9.25 – 5.75)2] + [2 × 0.20 ×8.044 × (5.75 −

2.25)2] = 1520.875in4ft�


ft� , it must be multiplied by 0.083334.

I = 1520.875 in4

ft� = 0.073345ft4ft�

3.3.4.1.3 Segment III

The figure below represents the bottom segment as a rectangle with a height of 15

inches.

31

Figure 21: Barrier Bottom Segment with Vertical Rebar in XY plane.

The only rebar contributing in strain energy absorption about the Z axis in the ZX

plane are vertical rebar and stirrups in the Y direction. The effect of stirrups is negligible.

It is the same as top and mid sections. The strain energy will be calculated for a unit

length of the barrier. The length of the section will be equal to one foot (12 inches), and

based on AASHTO Standard NEW JERSEY type barrier; the segment will be a rectangle

with 15 inches in height and 12 inches in wide. Figure 22 presents the modified section.

32

Figure 22: Plan View of Barrier Bottom Segment in ZX Plane



edge to the center of gravity of the top rebar is approximately 12.75 inches, and the area

of No.4 rebar is 0.20 in2. The total amount of steel area used in this section is,

(2 × 0.20) + (4 × 0.202

) = 0.8 in2



(3-19)


between the center of gravity and the bottom edge of the section of rebar. Therefore,

33


(3-19)

So that,

CG =(2 × 8.044 ×0.20 × 12.75)+(2 ×8.044 × 0.20 × 2.25)+(15 × 12 ×7.5)(2 ×8.044 × 0.20)+(2 ×8.044 × 0.20)+(15 ×12)

= 7.5 in.

Moment of inertia of the section is,

I = 𝑏 ℎ3

12 + (A d2)𝑐𝑜𝑛𝑐. + (A d2)𝑟𝑏𝑟. (3-20)

The distance from center of gravity of the concrete to the section center of gravity is zero,

so that, the second term of the equation above is equal to zero.

I= 12 × 153

12 + [2 × 0.20 ×8.044 × (12.75 – 7.5)2] + [2 × 0.20 ×8.044 × (7.5 − 2.25)2] =

3375in4ft�


ft� , it must be multiplied by0.083334.

I = 3375in4ft� = 0.16276ft4

ft�

3.3.4.1.4 Barrier Section Moment of Inertia

In order to calculate section moment of inertia, a weighted-mean method of three

segment’s moment of inertia is used. The top, middle and bottom segments’ height of the

barrier are 21, 10 and 3 inches respectively. So that,

34

I2 = 0.017026× 21 + 0.073345× 10 + 0.16276× 3 21 + 10 + 3

= 0.04644 ft4

ft�

3.3.4.2 Deck Overhang

Since the initial location of impact force is on the top and perpendicular to the

barrier’s face, in order to calculate the strain energy absorption of the deck overhang,

Impact force must be translated to the deck overhang. The SUPER POSITION method

indicated that by translating the impact force from the top of the barrier to midpoint of the

deck overhang thickness, a clockwise moment will be produced and is applied to the

midpoint of the deck overhang thickness. Its magnitude is barrier height H, plus half of

deck overhang thickness t, times impact force Ft. Figure 23 represents the SUPER

POSITION method procedure.

Figure 23: SUPER POSITION Method for Impact Force

The deck overhang moment of inertia I1 , must be calculated about the rotational

axis of the deck overhang which is the Z axis in ZY plane. The calculations for its

moment of inertia, are delineated on subsequent pages.

35

Figure 24: 3D View of Barrier and Deck

Since the strain energy is calculated for a unit length of the deck overhang, the

length of the section is equal to one foot (12 inches), so that, based on AASHTO

Standard bridge deck; the deck overhang segment will be a rectangle with 9 inches in

height and 12 inches in wide. Figure 25 presents the deck overhang section.

Figure 25: Deck Overhang Section in ZY Plane

36



edge to the center of gravity of the No.5 top rebar is 9 – 2.5 – 0.6252

= 6.187 inches; also,

the distance from the bottom edge to the center of gravity of the No.3 top rebar is 9 – 2.5

– 0.3752

= 6.312 inches, based on 2.5 inches cover from the top. The total amount of steel

area used in this section is,

(4 × 0.31) + (2 × 0.11) = 1.46 in2

Also the center of gravity of the section is,


(3-21)


between the center of gravity and the bottom edge of the section for rebar. Therefore,


(3-22)

So that,

CG =(2 × 8.044 × 0.31 × 6.187)+(2 ×8.044 × 0.11 × 6.312)+(2 ×8.044 ×0.31 × 2.312)+(9 × 12 ×4.5)(2 ×8.044 × 0.31)+(2 ×8.044 × 0.11)+(2 ×8.044 × 0.31)+(9 ×12)

= 5.85 in.

The moment of inertia of the section is

I = 𝑏 ℎ3

12 + (A d2)𝑐𝑜𝑛𝑐. + (A d2)𝑟𝑏𝑟. (3-23)

37

The distance from the center of gravity of the concrete and the section center of gravity is

zero. Therefore, the second term in the equation above is equal to zero.

I= 12 × 93

12 +[9× 12× (5.85 − 4.5)2] +[2 × 8.044 × 0.31× (6.187 − 5.85)2] + [2

× 8.044 × 0.11 × (6.312 − 5.85)2] + [2 × 8.044 × 0.31 × (5.85 − 2.312)2]

I = 1007.12in4ft�


ft� , it must be multiplied by0.083334. So,

I1 =1007.12in4ft� = 0.048569ft4

ft�

3.3.5 Displacement

3.3.5.1 Barrier

In order to calculate the horizontal displacement of the barrier caused by barrier

deflection Δ2, the barrier is assumed to act as a cantilever beam with vehicle impact force

on its span. The location of impact load depends on the location of the center of gravity

of the vehicle.

Δ2 = Fta2

6EIbarrier(3H − a) (3-24)

In the equation above, a represents the effective length of the barrier where force is

applied, E is the modulus of elasticity and H is the barrier height.

38

Figure 26: Barrier Elastic Deflection due to Vehicle Collision Force

Since the vehicle is Single Unit Truck and its center of gravity is 49 inches above

the ground, the collision force is applied to the top of the barrier with a height of 34

inches. So a = H; therefore,

Δ2 = FtH3

3EIbarrier = FtH3

3EI2 (3-25)

3.3.5.2 Deck Overhang

It has been assumed that the deck deflects vertically from its edge to the outer

edge of the nearest girder; however, the girders are considered to be rigid, so that the

deflection in vertical axis is negligible.

The figure below represents the horizontal deflection of the barrier due to vertical

deflection within the deck overhang.

39

Figure 27: Deck Overhang Vertical Deflection

Figure above 𝐼1 represents the deck overhang moment of inertia. 𝐼2 is the barrier moment

of inertia, l is the effective length of deck overhang, and M the produced moment which

causes the vertical deflection in the deck overhang.

In order to calculate Δ1 - the horizontal displacement of the barrier caused by deck

overhang deflection – the deck overhang is assumed to be cantilever a beam with a

moment produced by the Vehicle impact force on the free end. The magnitude of this

moment is calculated as M=𝐹𝑡a, and since the vehicle is Single Unit Truck, a is the

barrier height (H plus half of the deck overhang thickness t). Therefore, in order to

calculate the horizontal displacement of the barrier, the deflection angle of the deck

overhang must be calculated.

Θ=MlEI

= Ft(H + t

2) l

EI1 (3-26)

𝐼1 represents the moment of inertia of the deck overhang and l stands for the overhang

length.

40

Figure 28: Deck Overhang Deflection Configuration

Based on geometry, deck overhang deflection Angle θ, is equal to the barrier

displacement angle in the horizontal direction, so that,

Sinθ = Δ2

H+t2 (3-27)

Therefore Δ2 is:

Δ2 = (H + t2)Sinθ (3-28)

And by substituting θ,

Δ2 = (H + t2)Sin(

Ft(H + t2) l

EI1) (3-29)

Since deck overhang deflection angle θ is a small angle, Sinθ = θ,

Δ2 = �H + t2� Sin �

Ft(H + t2) l

EI1� =

�H+t2�

2Ftl

EI1 (3-30)

41

3.3.5.3 Superposition

The Total displacement in the horizontal axis was calculated by using the super

position method which is expressed as “The total displacement in horizontal direction

perpendicular to the barrier face (Δ), is equal to horizontal displacement of the barrier

perpendicular to its face before cracking (Δ2), plus horizontal displacement of the barrier

perpendicular to its face due to vertical displacement of the deck overhang ( Δ1).” Figure

29 represents the super position procedure.

Figure 29: Superposition Method for Barrier in Horizontal Deflection

Therefore, the total displacement of the barrier in horizontal axis Δ, is,

Δ =Δ2 + Δ1 = FtH3

3EI2 +

�H+t2�

2Ftl

EI1 (3-31)

3.3.6 Strain Energy Absorption due to Total Horizontal Displacement

The only unknown variable in the horizontal displacement equation is collision

force 𝐹𝑡, therefore the force-displacement curve is linear and the amount of strain energy

absorbed due to this displacement is the area under the curve and can be expressed as,

Wd=12

FtΔ (3-32)

42

By substituting Δ,

Wd=Ft2H3

6EI2+

Ft2l�H+t

2�2

2EI1 (3-33)

3.3.7 Moment Capacity of the Barrier

3.3.7.1 Vertical Moment Capacity, Mw

The horizontal reinforcement of the barrier provides moment capacity in the

vertical axis that is represented by Mw. Based on the vehicle impact strain rate, the

compressive strength of the concrete must be multiplied by the DIF factor.

𝑓𝑐 𝑛𝑒𝑤/ = 1.05 ∗ 4 = 4.2 ksi

Since the barrier thickness varies from top to bottom, in order to calculate 𝑀𝑤, the barrier

section must be divided into 3 segments.

3.3.7.1.1 Segment I

The positive and negative moment capacities of the top segment are

approximately equal and can be calculated as,

As =2- No. 3/s =2(0.11) = 0.22 in.2

davg =3 + 2.75 + 1.375

2= 3.56 in.

a= 𝐴𝑠𝑓𝑦

0.85 𝐷𝐼𝐹 𝑓𝑐/ 𝑏

= 0.22 × 600.85 × 1.05 × 4 × 21

=0.17607 in.

ΦMn1= ΦAsfy(d − a2) = 3.81916k.ft

43

Figure 30: Barrier Segment I in XY Plane. (After Barker, 2013)

3.3.7.1.2 Segment II

For this segment, the moment capacity is more complicated. Positive moment is,

As =1- No. 3s= 0.11in.2

dpos = 3.25+3.50= 6.75 in.

a= 0.11 × 600.85 × 1.05 × 4 × 10

= 0.18478 in.

ΦMnpos =1×0.11 × 60 × (6.75-0.1942

)= 3.66166 k.ft

Figure 31: Barrier Segment II in XY Plane. (After Barker, 2013)

44

And for negative moment,

dneg= 2.75+3.25= 6 in.

ΦMnneg =1×0.11 × 60 × (6.0-0.1942

)= 3.24916 k.ft

So that, the average value of positive and negative moment capacities is,

ΦMn2 = ΦMnpos+ΦMnneg

2 = 3.45541 k.ft

3.3.7.1.3 Segment III

For this segment the positive and negative moment capacity are equal and

As =1- No. 3s= 0.11in.2

d =9.5 + 2.75= 12.25 in.

a= 0.11 × 600.85 × 1.05 × 4 × 3

=0.61625 in.

Φ𝑀𝑛3=1×0.11 × 60 × (12.25-0.61622

)

= 6.56803 k.ft

Figure 32: Barrier Segment III in XY Plane. (After Barker, 2013)

The total moment capacity of the barrier about the vertical axis is the sum of moments in

all three segments.

45

Mw = ΦMn1+ ΦMn2+ ΦMn3= 13.84260 k.ft

3.3.7.2 Horizontal moment capacity, Mc

The vertical reinforcement of the barrier provides moment capacity in a horizontal

direction which is represented by Mc.Since the barrier thickness varies from top to

bottom, in order to calculate Mw, the barrier must be divided into 3 segments (the same

segments as in previous figures). The yield lines that cross the vertical reinforcement

produce tension; therefore, only negative moment capacity needs to be calculated.

Figure 33: Front View of Yield Line Failure Pattern. (After Calloway, 1993)

3.3.7.2.1 Segment I

As= 0.39 in.2

d = 7 − 2 − 0.25 = 4.75 in.

a= Asfy

0.85 DIF fc/ b

= 0.39 × 600.85 × 1.05 × 4 × 12

0.546218 in.

MC1= ΦAsfy(d − a2) = 1 × 0.39 × 60 × (4.75 - 0.546

2) = 8.7299 k.ft

46

3.3.7.2.2 Segment II and III

As= 0.39 in.2

d = 2 + 0.5 + 6 + 0.25 = 8.75in.

a= Asfy

0.85 DIF fc/ b

= 0.39 × 600.85 × 1.05 × 4 × 12

= 0.546218 in.

MC I+II= ΦAsfy(d − a2) = 1 × 0.39 × 60 × (8.75 - 0.546

2) = 16.5299 k.ft

The weighted-mean method was used in order to calculate the total moment capacity in

the horizontal direction of the barrier,

Mc = Mc1(21)+Mc2(10.0+3.0)34

= 11.7123 k.ft/ft

Also, in order to develop the horizontal moment capacity, 𝑀𝑐, through the whole length,

moment capacity along the Z-axis, must be multiplied by barrier length. So that,

Mc=(Mc/ft)* length (ft) (3-34)

3.3.7.3 Top Beam Moment Capacity

Since NEW JERSEY type barrier does not have a top beam, the moment capacity

along the top beam,𝑀𝑏, is equal to zero.

3.3.8 Internal Virtual Work along Yield Line, 𝐸𝑦𝑖𝑒𝑙𝑑

The internal virtual work along the yield lines is the sum of rotations and moment

capacities through the path within which they act. At the top segment of the wall, the

rotation θ is

47

θ ≈ tan θ =2δLc

(3-35)

Figure 34: Top View of Yield Line Failure Pattern. (After Calloway, 1993)

Assuming the negative and positive plastic moment capacities are equal, the internal

virtual work done by the top beam would be,

Ub=4MbΘ =8MbδLc

(3-36)

As mentined before, NEW JERSEY type barrier does not have a top beam, therefore

Ub = 0. The internal virtual work done by horizontal rebar can be expressed as,

Uw=4MwΘ =8MwδLc

(3-37)

The projection of displacement on the vertical path about the inclined yield line is 𝛿 𝐻� ;

therefore, the internal virtual work done by the vertical rebar 𝑈𝑐 can be expressed as

Uc=McLcδH

(3-38)

And the total internal virtual work along yield lines is

Wyield=8MbδLc

+ 8MwδLc

+ McLcδH

(3-39)

48

3.3.9 External Virtual work by Applied Load

Figure 35 presents the deformations of the barrier top beam with respect to its

original position. The shaded area represents the integral of this deformation through

which the distributed vehicle collision force 𝑤𝑡=𝐹𝑡/𝐿𝑡, acts. Based on that, for the total

displacement δ, the displacement x along the length is,

X=Lc−LtLc

δ (3-40)

Figure 35: External Virtual Work by the Impact Load. (After Calloway, 1993)

𝐿𝑡 is the length of distributed impact force 𝐹𝑡, and 𝐿𝑐 is the barrier critical length. The

shaded area can be presented as,

Area=12

(δ + X)Lt = δ LtLc

(Lc − Lt2

) (3-41)

The external virtual work done by distributed impact force wt, is expressed as,

W = wt(area)= FtΔLc

(Lc − Lt2

) (3-42)

49

3.3.10 Critical length of Yield-Line failure pattern, 𝐿𝑐

As already discussed at the beginning of the chapter, in modified yield line, deck

overhang and barrier act as individual cantilever beams with fixed ends bonded together

with a 90 degree angle at the deck overhang. In order to obtain more accurate results,

barrier and deck overhang lengths must be minimized to act more like cantilever beams.

Therefore, modified yield line approach becomes,

W - Wd= Wyield (3-43)

And by setting the above equation to zero, modified yield line approach can be expressed

as,

W - Wd - Wyield = 0 (3-44)

By substituting all the terms,

Ftβ2

− 8 MwβLc

− 8 MbβLc

− FtLtβ2 Lc

− Lc McβH

= 0 (3-45)

Where,

β = Ftb2H2 E I2

− Ftb3

6 E I2 + Ft b H L

2 𝐸 𝐼1 + Ft b l t

2 𝐸 𝐼1 (3-46)

In equations above,

𝐹𝑡 is the transverse impact force represented in kips;

𝑚1 is the vehicle mass presented in kips;

𝑚2 is the total mass of barrier presented in kips;

50

b is the vehicle center of gravity height represented in feet;

𝐿𝑡 is the length of the distributed vehicle collision force presented in feet;

𝐿𝑐 is the critical length of the barrier represented in feet;

l is the effective deck overhang length represented in feet;

H is the barrier height represented in feet;

Δ is the total displacement of the barrier in the horizontal direction caused by vehicle

impact and represented in feet;

t is the deck overhang thickness represented in feet;

E is the modulus of elasticity of the concrete represented in kips per square feet (ksf);

𝐼1 is the moment of inertia of the deck overhang section about rotation axis represented in

𝑓𝑡4;

𝐼2 is the moment of inertia of the barrier section about rotation axis represented in 𝑓𝑡4;

V is the combined mass velocity after the impact represented in mile per hour (mph);

𝑀𝑤 is the moment capacity of the barrier along vertical axis provided by reinforcement in

the horizontal direction presented in kips-ft;

𝑀𝑐 is the moment capacity of the barrier along horizontal axis provided by reinforcement

in the vertical direction represented in kips-ft;

𝑀𝑏 is the barrier top beam moment capacity represented in kips-ft;

θ is the angle of impact represented in radians;

Δt is the impact duration which the vehicle and barrier are considered as a unit mass with

the same velocity presented in second.

In order to find the critical length of the yield line pattern 𝐿𝑐, the modified yield

line equation must be solved for the impact force, 𝐹𝑡. So that,

51

Ft = − 2(McLc2+8HMw+8HMb)H (Lt−Lc)

(3-47)

The only unknown variable in this equation needed to determine the inclination of

yield line, α, is the critical length of the barrier. The value of the critical length of

barrier 𝐿𝑐,that minimizes the 𝐹𝑡 is determined by differentiating this equation, with

respect to the critical length of the barrier, and equating the result to zero.

dFtdLc

= 0 (3-48)

And by substituting the terms,

dFtdLc

= − 2 McLc2 + 16 H Mw + 16 H Mb

H ( Lt − Lc )2 − 4 LcMcH ( Lt −Lc )

= 0 (3-49)

This differentiation results in two quadratic results that must be solved for critical length

of barrier Lc,

Lc1 =LtMc+�Mc(McLt

2+8HMw+8HMb)

Mc (3-50)

And

Lc2 =LtMc−�Mc(McLt

2+8HMw+8HMb)

Mc (3-51)

and by substituting variables, the second equation for critical length always has a

negative value. Therefore;

Lc= Max {Lc1, Lc2} (3-52)

52

Figure 36: Yield Line Pattern Front View. (After Barker, 2013)

3.3.11 Nominal Railing Resistance to Transverse Loads, 𝑅𝑤

Minimum value of 𝐹𝑡 is obtained by substituting the calculated value of Lc in the

collision force equation. This value is denoted as nominal railing resistance to transverse

loads, Rw,

Min Ft= Rw (3-53)

According to AASHTO Bridge Specifications, nominal railing resistance to transverse

load Rw, must be greater than allowable collision force Ft, for the specified test level. The

following equation describes their relationship:

Rw > Ft (TL) (3-54)

53

Table 2: AASHTO Specifications Test levels Details. (AASHTO, A13.7.2-1.)

The nominal railing resistance Rw, has been transferred through a cold joint by

shear friction. Figure 37 represents the free-body diagram of this procedure. Assuming

the nominal railing resistance Rw, sheared out with the slope of 1 : 1 from the critical

length of the barrier Lc, the shear which is produced by the vehicle impact force 𝑉𝐶𝑇, or

Tensile force T, at the bottom of the barrier is calculated as,

VCT = T = RwLc+2H

(3-55)

54

Figure 37: Transferred Collision Force Between Barrier and Deck. (After Barker, 2013)

And based on AASHTO Bridge Specifications, the nominal shear resistance of the

interface plane 𝑉𝑛, is the minimum of,

Vn = min {�c Acv + µ �Avf Fy + Pc��, K1 DIF fc/Acv , K2 Acv} (3-56)

where:

Acv is the shear contact area which is:

15 × 12 = 180in.2

ft

55

Avf is the dowel area across shear plane which is 0.39 in.2

ft;

c is the cohesion factor which is based on [A5.8.4.2] 0.075 ksi;

fc/ is the compressive strength of the weaker concrete which is 4 kips per square inches

(ksi);

Fy is the yield strength of reinforcement which is 60 ksi;

Pcis the permanent compressive force which is equal to 0.320kips/ft;

µ is the friction factor based on [A.5.8.4.2] which is 0.6;

K1 is the fraction of concrete strength available to resist interface shear which as

specified in [A5.8.4.3] is 0.2;

K2 is the limit interface shear resistance factor based on [A5.8.4.3] which is 0.8 ksi

Factors c, µ, K1and K2 are for normal concrete placed against hardened concrete,

clean and without any laitance but not roughened. So that for one foot of design of

barrier,

K1 DIF fc/Acv= 2(1.05)(4)(180)=144 kips/ft

K2 Acv= 0.8(180)= 144 kips/ft

cAcv + µ �Avffy + Pc�=0.075(180) + 0.6 [0.39(60) + 0.320]=13.5+14.23=27.73 kips/ft

Therefore:

Vn= min {27.73, 144, 144} = 27.73 kips/ft

The nominal shear resistance must be greater than the shear produced by the truck

collision VCT. (Barker, 2013)

56

Vn> VCT (TL) (3-57)

Results are presented in chapter six.

57

4 ENERGY METHOD

4.1 Objective

The first objective of this chapter is to propose a method that involves the

conservation of energy from the impact. Based on that, the total energy interacting in the

system before the impact - Impact Severity - must be equal to the sum of total energy

absorbed by each component due to all deformations and displacements in the system and

all losses.

The second objective of this chapter is to calculate the energy absorption capacity

of barrier and cantilever overhang in elastic regions in order to compare them with the

amount of energy absorbed by each component in order to investigate whether the barrier

or overhang fails under TL-4 conditions or not, and if they do not fail, whether they

exceed their elastic limit and reach the plastic limit or not.

In this method it has been proposed that the moving vehicle energy prior to the

impact perpendicular to the barrier - Impact Severity - is less than the sum of the

following:

- strain energy absorbed by the barrier in elastic region of concrete Δ1

- strain energy absorbed by the barrier in the plastic region of concrete Δ2

- energy absorption by the vehicle due to its deformation Δ3

- strain energy absorbed by deck overhang in elastic region of concrete Δ4.

Therefore:

IS < (Δ1 or Δ2 )+Δ3+Δ4 (4-1)

58

4.2 Moving Vehicle Energy, SI

Since the vehicle has translational displacement due to constant velocity, the total

energy of the moving vehicle prior to the impact can be expressed as its overall kinetic

energy. This kinetic energy has 2 components to the barrier, perpendicular and parallel.

The only component that interacts with the barrier and deck overhang system is

the perpendicular component of the vehicle’s kinetic energy (IS). Based on TL-4, with 15

degree angle of crash, kinetic energy can be expressed as,

IS = 𝑤1 v2sin2θ2gc

(4-2)

In equation above, w is vehicle mass in pound, v is the initial velocity, 𝜃 is the impact

angle, and 𝑔𝑐 is the gravity acceleration.

59

Figure 38: Moving Vehicle Kinetic Energy

4.3 Barrier Strain Energy

4.3.1 Barrier Strain Energy Capacity in Elastic Region, 𝐸1

Figure 39 presents the stress-strain curve of concrete. As is shown, concrete in

tension starts with linear behavior followed by nonlinear behavior after cracks occur. The

red dot represents the change in phase of concrete. The strain energy capacity of

reinforced concrete in linear zone,E1, based on section properties can be expressed as,

E1= ∫ MB2 dx

2E I2

l0 =∫ Mc

2dx2E I2

l0 (4-3)

l is the barrier height, 𝑀𝐵 represents the moment capacity of the barrier from top view

which is the same as 𝑀𝑐, E is the modulus of elasticity of the concrete, and 𝐼2 is the

barrier moment of inertia.

If the total applied energy on the section exceeds strain energy capacity, concrete

starts to crack, which is the introduction to nonlinear behavior.

60

Figure 39: Stress-Strain Curve of Concrete

4.3.2 Absorbed Energy by the Barrier, 𝛥1

Conservation of momentum for the impact can be expressed as,

m1v1sinθ + m2v1 = (m1 + m2 ) V (4-4)

In the equation above, 𝑚1 represents the mass of the vehicle, 𝑣1 is the vehicle initial

velocity, 𝑚2 is the mass of barrier, and V represents the combined mass velocity where

the barrier and vehicle are considered as a unit mass.

In the conservation of momentum equation the barrier weight is 0.32 kips/ft and

the barrier length – assumed to be the same as vehicle length – is 26.6 feet, Therefore, the

barrier mass calculates as,

𝑚2 = 0.32 × 26.6 = 8.83 kips = 8837 lb

By calculating V from the conservation of momentum equation, the kinetic energy

absorbed by the barrier just after the impact can be expressed as,

Δ1 = Ebarrier, after impact = w2V2

2gc (4-5)

If this amount energy exceed the capacity of the section in elastic limit, the cracks start to

occur which is the introduction to the plastic region of the section.

4.4 Absorbed Energy by Barrier in Plastic Region, Δ2

As discussed before, concrete in tension has linear behavior with elastic deflection

followed by occurring cracks which is the introduction to nonlinear behavior. Δ2

61

represents the strain energy absorbed by the barrier in the nonlinear zone. Since

AASHTO Standard Bridge Specifications designs NEW JERSEY concrete barrier based

on a virtual failure pattern, which represents the cracks and failure modes in concrete, the

same approach can be used in this step as well.

Based on Yield Line approach, barrier strain energy in nonlinear zone can be expressed

as internal virtual work along yield lines,

Δ2= 8MbδLc

+8MwδLc

+McLcδH

(4-6)

In the equation above, Mb represents the top beam moment capacity, Mw represents the

vertical moment capacity of the wall provided by horizontal rebar, Mc is the horizontal

moment capacity of the wall provided by vertical rebar, δ is the horizontal displacement

of the barrier due to vehicle collision force, Lc is the critical length of the barrier, and H is

the barrier height.

Figure 40: External Virtual Work by the Impact Load. (After Calloway, 1993)

Since NEW JERSEY concrete barrier does not have a top beam, top beam

moment capacity is equal to zero. Also, based on AASHTO yield line design, the critical

62

length of the barrier Lc , is equal to 7.17 feet and Mw and Mc are 13.82 kips-ft and 11.69

kips-ft/ft (Barker, 2013). The only unknown variable in the above equation is horizontal

displacement of the barrier due to vehicle collision force and in order to calculate that,

LS-DYNA code, based on Finite Element Method, was used by setting equivalent

distributed force to the barrier. The perpendicular component of vehicle impact force to

the barrier is 54 kips which is distributed to a length of 3.5 feet. The position of this force

is the same position as the center of gravity of the single unit truck which is 49 inches

from the ground. Since the NEW JERSEY barrier height is 34 inches, the force will apply

to a defined area on top of the barrier. AASHTO Standard Bridge Specifications

recommend that the length of this distributed load is 3.5 feet and based on the truck

geometry the height of this area is 0.41 foot.

Initial assumption is that the barrier will not exceed the elastic limit and the total

energy absorbed by the barrier will be absorbed in the elastic region of section. By having

the elastic capacity and calculating the energy transferred to the barrier, it will be possible

to discuss whether Δ2 must be calculated or not.

4.5 Absorbed Energy by Vehicle Deformation, Δ3

There are two method proposed in this section in order to calculate absorbed

energy by vehicle deformation after the impact. The first method is using the National

Crash Analysis Center (NCAC) model provided in LS-DYNA code for TL-4 and the

second method has been proposed based on the conservation of energy equation for the

system before and after the impact. Figure 41 to Figure 43 represent the first method

which is vehicle deformation in LS-DYNA code based on National Crash Analysis

63

Center (NCAC) model for TL-4 which will be discussed in details in chapter 5 (NCAC,

2008).

Figure 41: Initial Vehicle Deformation

64

Figure 42: Final Vehicle Deformation

65

Figure 43: Vehicle Deformation Side View

The Second method proposes that vehicle deformation absorbs a large amount of

energy which means that this amount of energy will not be absorbed by the barrier or

deck overhang but will be absorbed directly by the deformation of the vehicle. Figure 44

presents a sample of vehicle deformation due to impact. This procedure is described as

follows:

66

Figure 44: Vehicle Deformation of TL-4. (After Sheikh, 2011)

By rewriting the kinetic energy equation for the vehicle perpendicular to the barrier after

the impact and taking into account the combined mass velocity,

Evehicle, after impact = w1V2

2gc (4-7)

Since the velocity of the vehicle perpendicular to the barrier after impact becomes zero,

this energy must be absorbed by the vehicle due to its deformation. Therefore,

Δ3 = Evehicle, after impact = w1V2

2gc (4-8)

4.6 Deck Overhang Strain Energy

4.6.1 Deck Overhang Strain Energy Capacity in Elastic Region, 𝐸4

The vertical displacement of the deck overhang presents strain energy absorption.

This energy cannot exceed more than deck overhang section capacity; otherwise the

67

deformation passes the elastic limit and starts to crack, which is the introduction to the

plastic limit. Therefore, the maximum moment capacity of the deck overhang must be

used in order to calculate the maximum allowable energy absorption by the deck

overhang. So that Δ4 can be expressed as,

E4= ∫ MD2 dx

2E I1

l0 (4-9)

l is the length of deck overhang, 𝑀𝐷 represents the moment capacity of the deck

overhang from side view, E is the modulus of elasticity of the concrete, and 𝐼1 is the deck

overhang moment of inertia.

As discussed before, AASHTO Standard Bridge Specifications indicates that,

based on TL-4, the impact load will be distributed to a length of 3.5 feet. Therefore, the

moment capacity and moment of inertia of the deck overhang must be calculated for 3.5

feet.

4.6.2 Absorbed Energy by Deck Overhang, 𝛥4

Based on conservation of energy for the barrier, deck overhang, and vehicle

system due to impact,

IS = (Δ1 or Δ2 ) +Δ3+Δ4 (4-10)

As already discussed, it has been assumed that the barrier does not exceed the elastic

limit and does not crack. Therefore:

Δ2 = 0 (4-11)

68

Therefore, the conservation of energy can be rewritten as,

IS = Δ1+Δ3+Δ4 (4-12)

so that, absorbed energy by deck overhang Δ4, can be expressed as,

Δ4 = IS – ( Δ1 +Δ3) (4-13)


69

5 LS-DYNA SIMULATION

The objective of this chapter is to develop a simulation tool by LS-DYNA code

for TL-4 and compare the results obtained from the simulation and analytical methods

(Work and energy methods) in order to verify the results of the analytical methods.

After testing this simulation tool experimentally, it can be used in future research

in order to validate different case scenarios and different test level aspects such as vehicle

speed, type and angle of impact.

5.1 Implementation of LS-DYNA

In order to simulate TL-4, it is important to use accurate data. Finite element

simulation of vehicle impact was modeled by LS DYNA code. This model represents the

NEW JERSEY barrier settled on the edge of an AASHTO standard bridge deck. The

simulation consists of two parts which are explained in details.

5.2 NCAC Model

The first model was obtained from the National Crash Analysis Center (NCAC)

website that represents TL-4. NCAC defined the barrier as JERSEY type barrier with

rigid material. The distance from the vehicle front to the impact point is defined as 6 feet.

The angle of impact is 15.45 degree and also the length of the barrier is defined as 118

feet. The NCAC model is based on SI units and has 302 time steps of 0.9 seconds each.

70

5.2.1 NCAC Single Unit Truck

The truck presented in NCAC model is a 18 kips Ford single unit truck 27.6 feet

long, 10.88 feet in height and 8 feet in width. The materials used in the truck are *MAT-

001ELASTIC, *MAT-009-NULL, *MAT-020-RIGID, *MAT-024-PIECEWISE-

LINEAR-PLASTICITY, *MAT-S01-DAMPER-VISCOUSE, and *MAT-SPRING-

NONLINEAR-ELASTIC. The element types used in the truck model are beam, discrete,

mass, seatbelt-accelerometer, shell, and solid elements, also the total number of elements

are 35400 consisting of 38939 nodes. The vehicle has a translational velocity of

22 977.856 mm/sec (51.4 miles per hour) as well as rotational velocity with the same

magnitude defined for wheels. The gravity force on the truck is defined based on gravity

acceleration, 9806 mm/s2 . The truck model consists of 151 individual parts with

individual sections. The air bag definition was based on SIMPLE-AIRBAG-MODEL and

6 different types of contacts were defined in the truck.

Figure 45: Single Unit Truck Top View

71

Figure 46: Single Unit Truck Side View

72

Figure 47: Single Unit Truck Front View

5.2.2 NCAC Rigid Barrier

The barrier presented in NCAC model is a JERSEY type barrier with dimensions

of 118 feet in length, 2.66 feet in height, 2 feet at the bottom, and 0.5 feet at the top

widths. The material used in barrier is *MAT-020-RIGID and the element type is

Belytschko-Tsay shell element; also the total number of elements are 15237 consisting of

15240 nodes. The NCAC barrier is settled on a rigid plane.

73

Figure 48: NCAC Barrier Top View

Figure 49: NCAC Barrier Front View

74

Figure 50: NCAC Barrier 3D View

5.2.3 Objective of NCAC Model

Since the only moving object in the NCAC model is the truck, and the barrier is

defined as a rigid body, the outputs of kinetic energy represent the kinetic energy of the

vehicle in every time step. Based on conservation of energy, the total amount of energy

absorbed by the vehicle deformation can be tracked by plotting the kinetic energy versus

time for the impact. Figure 51 to Figure 53 represent the NCAC TL-4 model.

75

Figure 51: NCAC Model Top View

76

Figure 52: NCAC Model 3D View

77

Figure 53: NCAC Model Side View

5.3 Proposed Model

The second model is created by LS-PrePost 4.2 software that consists of

AASHTO standard NEW JERSEY barrier installed on the standard bridge deck. Instead

of using the truck in this model, a simplified approach was used by using an equivalent

static force of single unit truck hitting the barrier based on TL-4. AASHTO indicates that

this equivalent force is 54 kips distributed in a 3.5 feet length perpendicular to the barrier.

It has been assumed that the force applies to an area with 0.41 feet height and 3.5 feet in

length with the duration of 0.175 seconds. The units used in the model are SI units.

78

The element type used for modeling the concrete was fully integrated quadratic 8

node solid element with nodal rotations with a length of approximately 25.4 mm (1 inch),

and the material model used for concrete was *MAT 159-CSCM-CONCRETE. This

material can reflect the cracks in concrete caused by plastic deformation. This material

was defined with the properties of normal strength concrete. MAT159 input keyword for

normal strength concrete was defined as,

The element type used for modeling the rebar was Hughes-Liu with cross section

integration beam elements, approximately 25.4 mm (1 inch) in length. The material

model used for rebar based on 60 ksi steel was *MAT-024-PEACEWISE-LINEAR-

PLASTICITY. The section dimensions for rebar No.3, No.4, and No.5 were defined

individually. Figure 54 represents the stress-strain curve for 60 ksiA36 steel (Getter,

2013). The horizontal axis represents the strain and the vertical axis represents the stress

in Newton/mm2. MAT024 input keyword was defined as,

And input keyword for a stress-strain curve of 60 ksi steel was defined as,

79

Figure 54: Stress-Strain Curve of 60 ksi Steel

Since the equivalent force is quasi static, the Strain rate parameters, C and P, are

set to zero.

The spacing of rebar varies in the section, therefore, in order to define the contact

between concrete and rebar and forcing them to act along each other, the elements’ length

was defined as 25.4 mm (1 inch) for both concrete and rebar. Then, all the nodes of

80

concrete and rebar with the distance of 12.7 mm were merged. This procedure forced

concrete and rebar to interact together.

5.3.1 Deck Overhang

Deck overhang was modeled based on AASHTO standard bridge deck. Figure 55

represents the section in detail. The thickness of overhang is 9 inches and the length

measured from the center of the outer column to the edge is 39 inches. Reinforcements of

the deck consist of alternating No.3 and No.5 at 7.5 inches at the top and No.5 at 9 inches

at the bottom, parallel to the section. Also longitudinal reinforcements were added No. 4

at 8 inches on top and No. 4 at 18 inches at the bottom.

5.3.2 NEW JERSEY Concrete Barrier

The barrier Section is 34 inches in height, with 6 inch widths on top and 15 inch

width at the bottom. Rebar consists of a total of seven No.3 longitudinal and three No.4

stirrups, 6 inches apart. Figure 55 represents the section.

81

Figure 55: Section Details. (After Barker, 2013)

82

Figure 56: Model Section

83

Figure 57: Model 3D Section

84

Figure 58: Section Element Formation

The length of the barrier and deck overhang was defined as 40 feet and the

equivalent impact force were defined at the middle of the barrier, 18.25 feet from

anchored. Since the barrier and deck will not be built in same time, the only connection

between barrier and deck will be stirrups. Therefore, the barrier and deck overhang were

modeled individually and only connected by two 2 stirrup “legs”. The effect of gravity in

this model was defined based on 9806 mm/s2( 932.2 lbm−ftlbt−sec2) gravity acceleration. Figure

59 to Figure 62 represent the model of the barrier and deck overhang system.

85

Figure 59: Model 3D view

Figure 60: Model Top View

86

Figure 61: Model Side View

87

Figure 62: Equivalent Distributed Force


88

6 RESULTS AND DISCUSSION

6.1 Work Method

Modified version of yield line analysis was presented in the first chapter based on

strain energy absorption of the barrier due to its horizontal displacement caused by

vertical displacement of deck overhang and strain energy absorption of the barrier in

horizontal direction before cracking.

The calculations of work method are based on TL-4, initial impact, and for a one-

foot strip of the barrier,

Crash angle θ= 15 degree = 15 × Π180

= 0.2617 Rad

Sinθ in radian is 0.2588 and weight of the barrier is 0.32 kips/ft. Therefore the combined

mass velocity for initial impact is,

V=m1v1 sin Θm1+m2

= 18 × 50 × 0.2588 18 + (0.32 × 3.5)

= 12.18 mph

Based on 0.1 second impact duration, the impact force Ft, can be written as,

Ft=m2V− m2v2

Δt = (0.32 × 12.715) − (0.32 × 0)

0.1 = 36.99 kips

The modulus of elasticity of normal concrete E, with 4000 pound per square inch

compression strength 𝑓𝑐/ , can be calculated as,

E = 57000 �𝑓𝑐

/

1000 = 57000 √4000

1000 = 519119.5 ksf

89

Based on a 1.05 dynamic increase factor; 34 inches or 2.83 feet (the TL-4

standard height of NEW JERSEY barrier) H; 0.04644 ft4 moment of inertia of the barrier

I2; 0.048569 ft4ft� moment of inertia of the deck overhang I1; 9 inches (0.75 foot)

standard deck thickness t; and 2.66 feet deck overhang length from the outer edge of the

girder l; the total displacement is,

Δ=Δ2 + Δ1 = 𝐹𝑡𝐻3

3𝐸𝐼2 +

�𝐻+𝑡2�

2𝐹𝑡𝑙

𝐸𝐼1 = 36.99 × 2.833

3 × 1.05 × 519119.5 × 0.04644 +

�2.83+0.752 �

236.99 × 2.66

519119.5 × 0.048569

= 0.005448 + 0.01616= 0.0216ft

By substituting 13.84k.ft vertical moment capacity of the barrier 𝑀𝑤; 11.71 k.ft

horizontal moment capacity of the barrier 𝑀𝑐; zero as top beam moment capacity 𝑀𝑏; and

3.5 feet distributed length of impact force l, critical length is,

Lc1 =LtMc + �Mc(McLt

2 + 8HMw + 8HMb)Mc

= (3.5 × 11.71) +�11.71 (11.71 × 3.52+8 × 2.83 × 13.84+8 × 2.83 × 0)11.71

= 9.74f

And,

Lc2 =LtMc − �Mc(McLt

2 + 8HMw + 8HMb)Mc

= (3.5 × 11.71) −�11.71 (11.71 × 3.52+8 × 2.83 × 13.84+8 × 2.83 × 0)11.71

= -2.74ft

And,

Lc= Max {Lc1, Lc2}= Max {9.74, −2.74}= 9.74 ft

90

Minimum value of 𝐹𝑡 is obtained by substituting the maximum value of Lc in the impact

force equation, which is expressed as nominal railing resistance to transverse loads, Rw,

MinFt= Rw (6-1)

Therefore,

Ft = Rw = − 2(McLc2+8HMw+8HMb)H (Lt−Lc)

= − 2(11.71 × 9.742+ 8 × 2.83 × 13.84 + 8 × 2.83 × 0)2.83 (3.5 − 9.74)

= 161 kips

The AASHTO Standard Bridge Specifications test-levels table presents 54 kips as

the required resistance to vehicle impact force for TL-4 , while modified yield line

method calculates the capacity of the barrier at 161 kips.

Table 3: AASHTO Specifications Test Levels Details

Transferred shear between barrier and deck overhang, VCT, is equal to tensile force, T,

and is equal to,

91

VCT = T = RwLc+2H

= 1619.74 + 2 × 2.83

= 10.44 kips/ft

Also based on AASHTO Standard Bridge Specifications the nominal shear resistance of

the interface is calculated as,

Vn = min {�c Acv + µ �Avf Fy + Pc��, K1 DIF fc/Acv , K2 Acv} (6-2)

And by substituting the variables,

�c Acv + µ �Avf Fy + Pc�� = [0.075 × 180 + 0.6 (0. 39 × 60 + 0.32)] = 27.732 kips/ft

K1 DIF fc/Acv = 0.2 × 1.05 × 4 × 180 = 151.2 kips/ft

K2 Acv = 0.8 × 180 = 144 kips/ft

Therefore:

Vn = min {27.732 ,151.2 , 144} = 27.732 kips/ft

Yield line method calculates the transferred shear between barrier and deck, VCT,

as 4.61 kips/ft , and the AASHTO Standard Bridge Specifications defines the nominal

shear resistance, Vn, as 27.73 kips/ft, while the modified yield line method calculates the

transferred shear between the barrier and deck, VCT, as 10.44 kips/ft. This value is greater

than 4.61 kips/ft but still within the limit of AASHTO Standard Bridge Specifications.

6.1.1 Results Compression

Table 4 presents the results regarding the yield lines and modified yield lines

methods based on TL-4.

92

Table 4: Work Method Results Comparison

As has been shown in the above table, the critical length calculated by the yield

line method is 7.17 feet while the modified yield line proved that the critical length is

9.74 feet.

Also the calculated railing resistance based on yield line method is 59.1 kips and

in modified yield line it has been proven that the actual railing resistance to impact force

is 161 kips. AASHTO requirements for railing resistance is 54 kips for railing resistance.

Therefore, the actual capacity is almost 3 times greater than AASHTO requirements

which leads to a very conservative design.

Yield line method calculates the shear transferred between barrier and deck

overhang at 4.61 kips/ft and in modified yield line method it has been shown that the

93

actual shear transferred between barrier and deck overhang is 10.44 kips/ft. AASHTO

requirements for shear transferred is 27.73 kips/ft. Therefore, the actual shear transfer is

greater than the one calculation based on yield line method, but it is still within AASHTO

limits.

6.2 Energy Method

The calculations of energy method are presented based on TL-4, initial, and

secondary impacts, also the design is based on impact length 3.5 feet.

6.2.1 Moving Vehicle Energy, IS

Total energy before impact can be expressed as moving vehicle kinetic energy -

Impact Severity - and is calculated as,

IS = w v2sin2θ2gc

= 18 ×1000 × 72.92 × 0.258 2 ×32.2

= 100689.4ft-lbf

v = 50 mph = 72.9 ft/sec

6.2.2 Barrier Strain Energy

6.2.2.1 Barrier Strain Energy Capacity, E1

Barrier elastic strain energy absorption due to its capacity is,

E1= ∫ MB2 dx

2E I2

l0 (6-3)

𝑀𝐵 is the moment capacity of the barrier provided by vertical bars; therefore, 𝑀𝐵 is the

same as 𝑀𝑐 and, for 3.5 feet strip of barrier, can be written as.

MB × 3.5 = Mc × 3.5 = 40.89 × 3.5 = 40.899 k-ft

94

Since the barrier thickness varies from top to bottom, therefore, in order to

calculate the strain energy, barrier must be divided into 3 segments. And by

substituting the variables in strain energy equation,

E1= ∫ Mc2dx

2E I2−1

0.250 + ∫ Mc

2dx2E I2−2

1.0830.25 + ∫ Mc

2dx2E I2−3

2.8331.083

=0.125 Mc2

E I2−1 + 0.5415 Mc

2

E I2−2 – 0.125 Mc

2

E I2−2 + 1.416 Mc

2

E I2−3 – 0.5415 Mc

2

E I2−3

= 0.125 ×40.892

519119.5 ×3.5 × 0.569 + 0.5415 ×40.892

519119.5 × 3.5 × 0.256 – 0.125 ×40.892

519119.5 × 3.5 × 0.256 + 1.416 ×40.892

519119.5 ×3.5 × 0.059 –

0.5415 ×40.892

519119.5 ×3.5 × 0.059

= 53223.19 ft-lbf

6.2.2.2 Absorbed Energy by Barrier in Elastic Region, Δ1

By rewriting the conservation of momentum for the secondary impact and

calculating the combined mass velocity V,

m1v1sinθ + m2v1 = (m1 + m2 ) V (6-4)

Therefore:

V= m1v1 sinθm1+m2

= 18 ×1000 × Sin(Rad 15) ( 18 ×1000)+( 0.32 ×27.61 ×1000)

= 12.73 ft/sec

so that, the kinetic energy absorbed by the barrier just after the impact can be calculatee

as,

Δ1= EBarrier, after impact = w2V2

2gc = 0.32 ×27.6 ×1000 ×12.732

2 ×32.2 = 22237.89 ft-lbf

95

And since Δ1 is less than and about 40 percent of overall capacity of the barrier (E1), the

barrier will not reach the allowable capacity and will not exceed the elastic limit.

Therefore, there will not be any crack or failure in the barrier caused by impact load.

6.2.2.3 Absorbed Energy by Barrier in Plastic Region, Δ2

Since AASHTO Standard Bridge Specifications designs the NEW JERSEY

concrete barrier based on yield line pattern which represents the cracks and failure modes

in concrete. The same approach can be used in this step as well.

Δ2= Wyield line= 8MbδLc

+8MwδLc

+McLcδH

(6-5)

However, since the barrier does not exceed the elastic limit of the section, Δ2 is equal to

zero .

If the overall capacity (neglecting the effect of plastic limit) is less than TL-4

energy, then by obtaining the horizontal displacement of the barrier from LS DYNA

simulation and substituting the variables in equation above, Δ3 can be expressed as,

Δ2= 8MbδLc

+8MwδLc

+McLcδH

= 8 ×0 × δ7.17

+8 ×13.82 × δ7.17

+11.68 ×7.17 × δ2.83

= 8 ×13.82 × δ7.17

+11.68 ×7.17 × δ2.83

=15.41 δ + 29.59 δ = 45 δft-lbf

6.2.3 Absorbed Energy by Vehicle, 𝛥3

As it has been explained in chapter five, there are two methods represented in this

research in order to calculate the amount of energy absorbed by the vehicle deformation

after impact. First, the conservation of energy equation and is used to calculate the energy

96

absorbed by vehicle deformation and the second method is designed to obtain data from

the NCAC model.

The first case scenario implies that since the only motion of a vehicle after the

impact is parallel to the barrier and does not have motion perpendicular to said barrier

anymore, perpendicular kinetic energy of the vehicle just after the impact must be

absorbed by its deformation. Therefore, the perpendicular kinetic energy of the vehicle

after the impact must be equal to the absorbed energy by the vehicle due to its

deformation. This deformation energy can be calculated as,

Δ3−1 =EVehicle, after impact = w1V2

2gc = 18 ×1000 ×12.732

2 ×32.2 = 45296.09 ft-lbf

Δ3−1= 45% of Vehicle Kinetic Energy

Second case scenario is to calculate the absorbed energy by the vehicle

deformation and is based on the NCAC model. Appendix A represents the output data

obtained from the kinetic energy of the vehicle for different time steps from the NCAC

model. Based on the simulation, the impact starts from the 17th time step. Therefore, by

dividing the kinetic energy of this time step to the initial steady state kinetic energy

before impact, the fraction of remaining kinetic energy in the vehicle, with respect to total

energy before impact, is obtained. The same procedure must be done for all the time steps

after the impact. The amount of absorbed energy by the vehicle deformation can be

obtained by subtracting this fraction from one, and the total percentage of absorbed

energy due to vehicle deformation in each time step can be obtained by multiplying each

result by 100. Figure 63 presents the change in kinetic energy of the vehicle during the

simulation.

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The percentage of kinetic energy absorbed by the vehicle deformation after

impact is the final fraction of absorbed energy by said vehicle,

Δ3−2 = 56 % of Vehicle Kinetic Energy

Figure 63: Change in Kinetic Energy of the Vehicle Based on TL-4

Total absorbed energy by the vehicle deformation is expressed as the minimum of

absorbed energy based on kinetic energy and absorbed energy based on the NCAC

model.

Δ4= min { Δ3−1, Δ3−2 }= min { 45%, 56% }= 45% = 45296.09 ft-lbf

6.2.4 Deck Overhang Strain Energy

6.2.4.1 Deck Overhang Strain Energy Capacity, 𝐸4

Deck overhang strain energy absorption due to vertical displacement based on its

capacity is,

98

E4= ∫ MD2 dx

2E I1

l0 (6-6)

Figure 64: 1 Foot Strip of Deck Overhang Section in ZY Plane

And the moment capacity of the deck overhang for 1 foot strip calculates as,

As = 2 × 0.11 + 2 × 0.31 = 0.84 in2

d = 9 – 2.5 – 0.492

= 6.255 in

a= Asfy

0.85 fc/ b

= 0.84 × 600.85 × 4 × 12

= 1.235 in

Mn=Asfy(d − a2) = 0.84 × 60 × (6.255 – 1.235

2 ) = 23.67 k-ft/ft

Therefore, the factored capacity for the 3.5 foot strip is,

ΦMn = 0.836 × 23.67 × 3.5 = 69.278 kips-ft = 69278.557 ft-lbf

And by substituting all the variables in the strain energy equation,

99

E4= ∫ MD2 dx

2E I1

l0 = MD

2 l2E I1

= 69278.552× 2.6672 ×519119.5 ×3.5× 0.1699

= 72516.311 ft-lbf

6.2.4.2 Absorbed Energy by Deck Overhang in Elastic Region, Δ4

Based on conservation of energy being low for the barrier, deck overhang, and vehicle

system due to impact,

IS = ( Δ1or Δ2) + Δ3+ Δ4 (6-7)

And since the barrier does not exceed the elastic limit of the section,

Δ2 = 0 (6-8)

So that, the conservation of energy can be rewritten as,

IS = Δ1+ Δ3+ Δ4 (6-9)

Therefore the absorbed energy by deck overhang Δ4, can be expressed as,

Δ4 = IS – ( Δ1 +Δ3) = 100689.4 – (22237.89 + 45296.09) = 33142.86 ft-lbf

Since Δ4 is less than and almost 45 percent of total capacity of deck overhang

(E4), the deck overhang will not reach the allowable capacity and will not exceed the

elastic limit. Therefore, the deck overhang will not fail or crack under impact load.

6.3 Results from Proposed Model

In Work and Energy methods chapters, it has been discussed that NEW JERSEY

concrete barrier designed based on TL-4 is conservative and over designed. The same

100

results were obtained by applying the equivalent force of TL-4 to the barrier. Neither

barrier nor deck overhang exceeded the elastic limit, due to the strain energy absorbed by

the barrier and deck overhang within the limits of the elastic region. As already expected,

the maximum horizontal displacement occurs in the barrier at midpoint on top. Figure 65

presents the displacement verses time for this point.

Figure 65: Maximum Displacement of Barrier

Figure 66: Barrier Maximum Displacement with Scale of 50

101

Displacement and time are in millimeters and seconds respectively. As is shown

the maximum displacement in the horizontal direction is almost linear up to 0.11 seconds

and reaches 12 millimeters. Also, from 0.11 seconds the change in displacement is less

than 1 millimeter.

Maximum displacement in vertical direction occurs in the deck overhang at mid-

span under the load. Figure 67 presents the displacement verses time for this point.

Figure 67: Maximum Displacement of Deck Overhang

102

Figure 68: Deck Overhang Maximum Displacement with Scale of 50

As it is shown, the maximum displacement in vertical direction is almost linear up

to 0.11 seconds and reaches 1.62 millimeters. Also from 0.11 seconds the change in

displacement is less than 0.5 millimeters.

Stress distribution in barrier has an almost linear relationship with time up to 0.11

seconds and the position of maximum stress produced in the barrier is on the tension side

under the load at 0.11 second with magnitude of 28.7 MPa. Figure 69 presents barrier

maximum effective stress versus time. Stress and time are in MPa and seconds

respectively. As it is shown, the maximum stress is almost linear up to 0.11 seconds and

reaches 28.7 MPa. Also from 0.11 seconds the change in stress is less than two MPa.

Since the system was modeled based on criteria of normal strength concrete with 30 MPa

maximum stress, the concrete in barrier will not crack and or fail under the applied load.

103

Figure 69: Maximum Effective Stress in Barrier

104

Figure 70: 3D View of Barrier Maximum Effective Stress

Stress distribution in the deck overhang has a linear relationship with time up to

0.11 seconds and the position of maximum stress produced in the deck overhang is at

midpoint in the same position as maximum deflection at 0.11 seconds with a magnitude

of almost 19.5 MPa. Figure 71 presents deck overhang maximum effective stress versus

time. Stress and time are in MPa and seconds respectively. As it is shown the maximum

stress is almost linear up to 0.11 second and reaches 19.5 MPa. Also from 0.11 seconds

the change in stress is less than one MPa. Since the system was modeled based on criteria

of normal strength concrete with 30 MPa maximum stress, the concrete in deck overhang

will not crack and or fail under the applied load.

105

Figure 71: Maximum Effective Stress in Deck Overhang

106

Figure 72: Bottom View of Deck Overhang, Maximum Effective Stress

Maximum axial force resultant in rebar has a linear relationship with time up to

0.11 seconds, and the positions of maximum force resultant in rebar are in stirrups and

vertical rebar of barrier on the tension side, and rebar on top of deck overhang at

midpoint of the model at 0.11 seconds. Figure 73 and Figure 74 present rebar axial force

resultant. As it is shown, the maximum axial force resultant is 36180 Newton in rebar

No.4 and No.5 and 17430 Newton in rebar No.3; therefore, the stress produced in these

rebar will be,

No. 3 => Ơ = F

Area =

1743070.96

= 245.6 MPa

No. 4 => Ơ = F

Area =

36180129.03

= 280.39 MPa

No. 5 => Ơ = F

Area =

36180199.99

= 180.9 MPa

107

Since the rebar was modeled based on criteria of 60 Ksi steel which means the

yield stress of rebar is 420 MPa, the critical rebar of the section (No.3, No.4 and No.5)

will not yield and or fail under the applied load, therefore the section will not reach the

yield point.

Figure 73: Axial Force Resultant Distribution

108

Figure 74: Rebar Front View

As was expected, the deck overhang and barrier system did not exceed the elastic

limit of concrete; therefore, concrete will not crack under TL-4 impact load. Also none of

the rebar resulted in yielding or failure.

In overall view, the system displayed linear behavior in displacement and stress

distribution up to 0.11 seconds, and after that, the change in displacement had a

maximum one millimeter tolerance and the change in stress distribution has a maximum

two MPa tolerance.

The effective impact time is 0.11 seconds; therefore, the estimation of effective

impact time in the work method chapter – which was 0.1 – is verified.

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7 SUMMARY AND CONCLSION

There were 2 methods presented in this research - work and energy methods - in

order to prove that the AASHTO standard NEW JERSEY concrete barrier is very

conservatively designed. This procedure was followed by the simulation tool in LS-

DYNA application in order to validate the data gathered from both work and energy

methods.

7.1 Work Method

The first method was the work method presented in chapter 3. This method was

proposed based on yield line equation with taking into account the contribution of deck

overhang deflection in vehicle impact. In this method it has been implied that external

virtual work along the yield line minus work done by the deck overhang deflection is

equal to the internal virtual work done by the moment resistance along the yield lines,

W − Wd = W yield (7-1)

The work method procedure started by using the conservation of momentum for

the impact and then calculated the combined mass velocity of the system right after the

impact and followed that by calculating the change in momentum of the barrier (impulse)

in order to obtain an equation for impact force Ft. The next step was to calculate the

moment capacity of the barrier section with the contribution of the vehicle impact strain

rate and the moment of inertia of the total section in ZX plane, about rotational axis Z.

Next the procedure used the super position method for calculating the total horizontal

displacement of the barrier with accounting for the barrier deflection in the elastic region,

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and as well, overhang deflection in the vertical axis in the elastic region. Then the work

done by total displacement was calculated with the assumption that the barrier

displacement is linear. Next the internal virtual work along the yield lines in the barrier

and external virtual work done by applied load, was calculated. Equation 7-1 was

rewritten based on the impact force and differentiated with respect to critical length in

order to calculate the maximum critical length. Next the obtained critical length in

equation 7-1 was substituted to obtain the nominal railing resistance to transverse loads

of the barrier and compared to both the yield line method and AASHTO requirements.

The next step was to calculate the shear transferred between the barrier and deck

overhang and to compare the result with both yield line method and AASHTO

specifications.

7.1.1 Work Method Conclusions

As it has been shown in the table below, the critical length calculated by the yield

line method is 7.17 feet while the modified yield line proved that critical length is equal

to 9.74 feet.

Also the calculated railing resistance, based on yield line method is 59.1 kips and

in modified yield line it has been proven that the actual railing resistance to impact force

is 161 kips, while AASHTO requirements for railing resistance is 54 kips. The actual

resistance capacity is almost 3 times greater than AASHTO requirements indicating a

possibility of a conservative design.

The yield line method calculates the shear transferred between barrier and deck

overhang as 4.61 kips/ft while in modified yield line method, the actual shear transferred

111

between barrier and deck overhang is 10.44 kips/ft. AASHTO requirements for shear

transferred is 27.73 kips/ft. Therefore, the actual shear transfer is greater than the one

calculated by yield line method but still within AASHTO limits.

Table 5: Work Method Results Comparison

7.2 Energy Method

The second method was the energy method presented in chapter 4. The first

objective of this method was to propose a model illustrating conservation of energy

during impact. This method demands that the total energy interacts within the system

before the impact - Impact Severity - must be equal to the sum of the total energy

112

absorbed by each component including all deformations and displacements in the system

and all losses.

The second objective of this chapter is to calculate the energy absorption capacity

of the barrier and cantilever overhang in the elastic region in order to compare them with

the amount of energy absorbed by each component and to investigate whether or not

barrier or overhang fail under the Test Level 4 (TL-4) condition, and, if they do not fail,

whether or not they exceed their elastic limit and reach the plastic limit.

In this method it has been proposed that the moving vehicles energy,

perpendicular to the barrier prior to the impact - Impact Severity (IS) -, is less than the

sum of strain energy absorbed by barrier in the elastic region of concrete Δ1or the strain

energy absorbed by barrier in the plastic region of concrete Δ2, the energy absorption by

the vehicle due to its deformation Δ3, and the strain energy absorbed by the deck

overhang in the elastic region of concrete Δ4. Therefore,

IS < (Δ1 or Δ2) +Δ3+Δ4 (7-2)

7.2.1 Energy Method Conclusions

The proposed energy method indicates that under AASHTO TL-4 conditions,

barrier and deck overhang energy absorption will not exceed their capacities in elastic

limit, and therefore, there will not be any failure in barrier and overhang systems. Also, it

has been shown that the barrier will absorb about 40 percent of its overall capacity in its

elastic region, and the deck overhang will absorb almost 45 percent of its total capacity in

its elastic region. Also, it was shown that a large amount of energy is going to be

absorbed by vehicle deformation.

113

There were two methods presented in order to calculate energy absorption by

vehicle deformation - using conservation of energy and NCAC model for TL-4 -. The

conservation of energy method calculated that 46 percent of vehicle impact energy

perpendicular to the barrier is going to be absorbed by vehicle deformation, and the

NCAC model proposed that 56 percent of vehicle impact energy perpendicular to the

barrier is going to absorbed by the vehicle deformation.

7.3 LS-DYNA Model

The objective of the LS-DYNA model was to develop a simulation tool by LS-

DYNA code for TL-4 and to compare the results obtained from the simulation with the

analytical methods (Work and energy methods) in order to verify the results of the

analytical methods.

This simulation tool can be used in future research in order to validate different

case scenarios and different test level aspects, such as vehicle speed, type and angle of

impact.

7.3.1 LS-DYNA Model Conclusions

As was expected, the deck overhang and barrier system did not exceed the elastic

limit of concrete; therefore, concrete will not crack under TL-4 simplified impact load

and also none of the rebar resulted in yielding or failure under TL-4 simplified impact

load.

In overall view, the system displayed linear behavior in displacement and stress

distribution up to 0.11 seconds, and after that, the change in displacement had a

114

maximum one millimeter tolerance and the change in stress distribution has a maximum

two MPa tolerance.

The effective impact time is 0.11 seconds; therefore, the estimation of effective

impact time in the work method chapter – which was 0.1 – is verified.

7.4 Recommendations for future studies

This study indicated that NEW JERSEY concrete barrier may be conservatively

over designed based on AASHTO yield line design. Therefore, there is an opportunity to

reduce the section capacity in order to satisfy the criteria of yield line design (occurring

failure pattern and plastic region in barrier) and to reach the barrier and overhang strain

energy capacities in the elastic region.

By reducing the section capacity, a more economical design can be produced. For

instance, it is possible to achieve lighter guardrails set on the bridge deck which would

lead to reduction in the design load of bridge decks and a lighter design for bridge decks

and piers.

Author’s recommendations for future studies

1. Running field tests with different aspects of test levels and impacts in order to

verify simulation results and validating the data obtained from analytical methods.

One of the current research projects in the Civil and Mechanical Engineering

Departments of the University of Alabama at Birmingham is to perform field tests

with different aspects of test levels and impacts and to verify the results obtained

from simulations.

115

2. Reducing the section capacity and redesigning the section. Since the capacity is

almost three times more than the required capacity, redesigning could result in

reduced capital out lay and production costs.

3. Running field tests with different test levels and impacts with a new design in

order to come up with a new verified design for bridge rails, that is more

economical while still maintaining safety standards.

4. By applying the new verified design on bridge rails, it would be possible to

reduce the design loads (dead load) on the bridge decks and therefore, the bridge

piers. The next step would be to design lighter decks and bridge piers due to the

reduction in design loads. This procedure could be very cost efficient for future

bridge designs.

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LIST OF REFERENCES

1 AASHTO (2010). LRFD Bridge Design Specifications, 5th ed., American

Association of State Highway and Transportation Officials, Washington, DC. 2 Barker, R., & Puckett, J. (2013). Concrete Barrier Strength and Deck Design.

In “Design of highway bridges an LRFD approach (3rd ed.).” Hoboken, N.J.: John Wiley & Sons.

3 Bligh Roger P, Menges Wanda L. (2010) “Mash Test 3-11 on the Texas T101

Bridge Rail.” Report submitted to Texas Transportation Institute under report No. FHWA/TX-11/9-1002-1

4 Calloway, B. R. (1993). Yield Line Analysis of an AASHTO NEW JERSEY

Concrete Parapet Wall, M.S. Thesis, Virginia Polytechnic Institute and State University, Blacksburg, VA.

5 Hirsch, T. J. (1978). “Analytical Evaluation of Texas Bridge Rails to Contain

Buses and Trucks,” Research Report 230-2, August, Texas Transportation Institute, Texas A&M University, College Station, TX.

6 National Crash Analysis Center. (2008, November 3). Finite Element Model

Archive. Retrieved September 9, 2014, from http://www.ncac.gwu.edu/vml/models.html

7 Pajak Malgorzata. (2011) “The Influence of the Strain Rate on the Strength of Concrete Taking into Account Experimental Techniques.” Research Report 3/2011, The Silesian University of Technology.

8 Reid John D., Rosenbaugh Scott K., Haskall Jason A., Bielenberg Robert E.,

Polivka Karla A., Faller Ronald K., Allison Erin M., Rohde John R., Sicking Dean L. (2008) “Development of a stand-alone concrete bridge pier protection system.” Report submitted to the Midwest States’ Regional Pooled Fund under report No. TRP-03-190-08

9 Sheikh Nauman M., Bligh Roger P, Menges Wanda L. (2007) “Crash testing and

Evaluation of F-Shape Barriers on Slopes.” Report submitted to Texas Transportation Institute under report No. FHWA/TX-08/0-5210-3

10 Sheikh Nauman M., Bligh Roger P, Menges Wanda L. (2011) “Determination of

Minimum Height and Lateral Design Load for Mash Test Level 4 Bridge Rails.” Report submitted to Texas Transportation Institute under report No. FHWA/TX-12/9-1002-5

http://www.ncac.gwu.edu/vml/models.html

117

APPENDIX A

NCAC MODEL KINETIC ENERGY OUTPUTS

118

T. S. Time Kinetic Energy Remained Energy Absorbed by Vehicle 1 0.00000000 2100231000 1.000000000 0.000000000 2 0.00499694 2100981000 1.000357104 -0.000357104 3 0.00999698 2101314000 1.000515658 -0.000515658 4 0.01499702 2101704000 1.000701351 -0.000701351 5 0.01999706 2102100000 1.000889902 -0.000889902 6 0.02499710 2102051000 1.000866571 -0.000866571 7 0.02999714 2102205000 1.000939897 -0.000939897 8 0.03499718 2102403000 1.001034172 -0.001034172 9 0.03999722 2102518000 1.001088928 -0.001088928 10 0.04499726 2102400000 1.001032744 -0.001032744 11 0.04999730 2102331000 1.000999890 -0.000999890 12 0.05499734 2102326000 1.000997509 -0.000997509 13 0.05999738 2102383000 1.001024649 -0.001024649 14 0.06499743 2102420000 1.001042266 -0.001042266 15 0.06999747 2102295000 1.000982749 -0.000982749 16 0.07499751 2101137000 1.000431381 -0.000431381 17 0.07999755 2098751000 0.999295316 0.000704684 18 0.08499759 2096887000 0.998407794 0.001592206 19 0.08999763 2094304000 0.997177929 0.002822071 20 0.09499767 2092164000 0.996158994 0.003841006 21 0.09999771 2090384000 0.995311468 0.004688532 22 0.10499770 2087784000 0.994073509 0.005926491 23 0.10999780 2085004000 0.992749845 0.007250155 24 0.11499780 2079653000 0.990202030 0.009797970 25 0.11999790 2071880000 0.986501009 0.013498991 26 0.12499790 2066587000 0.983980810 0.016019190 27 0.12999790 2062217000 0.981900086 0.018099914 28 0.13499800 2056944000 0.979389410 0.020610590 29 0.13999800 2049778000 0.975977404 0.024022596 30 0.14499810 2043199000 0.972844892 0.027155108 31 0.14999810 2035138000 0.969006743 0.030993257 32 0.15499810 2021018000 0.962283673 0.037716327 33 0.15999820 2009412000 0.956757614 0.043242386 34 0.16499820 2002613000 0.953520351 0.046479649 35 0.16999830 1997126000 0.950907781 0.049092219 36 0.17499830 1989612000 0.947330079 0.052669921 37 0.17999830 1978952000 0.942254447 0.057745553 38 0.18499840 1973601000 0.939706632 0.060293368 39 0.18999840 1968335000 0.937199289 0.062800711 40 0.19499850 1963357000 0.934829074 0.065170926 41 0.19999850 1957767000 0.932167462 0.067832538 42 0.20499850 1951387000 0.929129700 0.070870300 43 0.20999860 1945550000 0.926350482 0.073649518 44 0.21499860 1940644000 0.924014549 0.075985451 45 0.21999870 1935649000 0.921636239 0.078363761 46 0.22499870 1930817000 0.919335540 0.080664460 47 0.22999870 1924987000 0.916559655 0.083440345 48 0.23499880 1920961000 0.914642723 0.085357277 49 0.23999880 1917988000 0.913227164 0.086772836 50 0.24499890 1914022000 0.911338800 0.088661200 51 0.2499989 1910323000 0.909577566 0.090422434 52 0.2549990 1906633000 0.907820616 0.092179384 53 0.2599990 1903010000 0.906095568 0.093904432

119

54 0.2649990 1899943000 0.904635252 0.095364748 55 0.2699991 1897401000 0.903424909 0.096575091 56 0.2749991 1894514000 0.902050298 0.097949702 57 0.2799991 1892544000 0.901112306 0.098887694 58 0.2849992 1890512000 0.900144794 0.099855206 59 0.2899992 1888743000 0.899302505 0.100697495 60 0.2949993 1886783000 0.898369275 0.101630725 61 0.2999993 1885044000 0.897541270 0.102458730 62 0.3049994 1882710000 0.896429964 0.103570036 63 0.3099994 1880588000 0.895419599 0.104580401 64 0.3149994 1878123000 0.894245919 0.105754081 65 0.3199995 1876744000 0.893589324 0.106410676 66 0.3249995 1875231000 0.892868927 0.107131073 67 0.3299995 1873914000 0.892241853 0.107758147 68 0.3349996 1872358000 0.891500983 0.108499017 69 0.3399996 1870737000 0.890729163 0.109270837 70 0.3449997 1869358000 0.890072568 0.109927432 71 0.3499997 1867758000 0.889310747 0.110689253 72 0.3549998 1866007000 0.888477029 0.111522971 73 0.3599998 1864235000 0.887633313 0.112366687 74 0.3649967 1861997000 0.886567716 0.113432284 75 0.3699968 1857713000 0.884527940 0.115472060 76 0.3749968 1853191000 0.882374844 0.117625156 77 0.3799968 1848536000 0.880158421 0.119841579 78 0.3849969 1844789000 0.878374331 0.121625669 79 0.3899969 1840794000 0.876472159 0.123527841 80 0.3949970 1837492000 0.874899951 0.125100049 81 0.3999970 1835026000 0.873725795 0.126274205 82 0.4049971 1832112000 0.872338328 0.127661672 83 0.4099971 1830024000 0.871344152 0.128655848 84 0.4149971 1827698000 0.870236655 0.129763345 85 0.4199972 1825221000 0.869057261 0.130942739 86 0.4249972 1823673000 0.868320199 0.131679801 87 0.4299972 1822796000 0.867902626 0.132097374 88 0.4349973 1819954000 0.866549441 0.133450559 89 0.4399973 1816141000 0.864733927 0.135266073 90 0.4449974 1813661000 0.863553104 0.136446896 91 0.4499974 1812437000 0.862970311 0.137029689 92 0.4549975 1809877000 0.861751398 0.138248602 93 0.4599975 1805041000 0.859448794 0.140551206 94 0.4649975 1799494000 0.856807656 0.143192344 95 0.4699976 1791603000 0.853050450 0.146949550 96 0.4749976 1782330000 0.848635222 0.151364778 97 0.4799976 1774678000 0.844991813 0.155008187 98 0.4849977 1764457000 0.840125205 0.159874795 99 0.4899977 1756396000 0.836287056 0.163712944 100 0.4949978 1748879000 0.832707926 0.167292074 101 0.4999978 1742964000 0.829891569 0.170108431 102 0.5049978 1735572000 0.826371956 0.173628044 103 0.5099979 1730323000 0.823872707 0.176127293 104 0.5149979 1724396000 0.821050637 0.178949363 105 0.5199980 1720927000 0.819398914 0.180601086 106 0.5249980 1717182000 0.817615777 0.182384223 107 0.5299981 1713849000 0.816028808 0.183971192

120

108 0.5349981 1709968000 0.814180916 0.185819084 109 0.5399981 1707141000 0.812834874 0.187165126 110 0.5449982 1705145000 0.811884502 0.188115498 111 0.5499982 1702935000 0.810832237 0.189167763 112 0.5549982 1700141000 0.809501907 0.190498093 113 0.5599983 1699131000 0.809021008 0.190978992 114 0.5649983 1698284000 0.808617719 0.191382281 115 0.5699984 1696615000 0.807823044 0.192176956 116 0.5749984 1695366000 0.807228348 0.192771652 117 0.5799984 1694275000 0.806708881 0.193291119 118 0.5849985 1693551000 0.806364157 0.193635843 119 0.5899985 1692882000 0.806045621 0.193954379 120 0.5949985 1692481000 0.805854689 0.194145311 121 0.5999986 1691818000 0.805539010 0.194460990 122 0.6049986 1690944000 0.805122865 0.194877135 123 0.6099987 1690092000 0.804717195 0.195282805 124 0.6149988 1689189000 0.804287243 0.195712757 125 0.6199988 1688448000 0.803934424 0.196065576 126 0.6249988 1688079000 0.803758729 0.196241271 127 0.6299989 1687417000 0.803443526 0.196556474 128 0.6349989 1686855000 0.803175936 0.196824064 129 0.6399989 1685453000 0.802508391 0.197491609 130 0.6449990 1682303000 0.801008556 0.198991444 131 0.6499990 1677212000 0.798584537 0.201415463 132 0.6549991 1673798000 0.796959001 0.203040999 133 0.6599991 1672025000 0.796114808 0.203885192 134 0.6649991 1669218000 0.794778289 0.205221711 135 0.6699992 1664311000 0.792441879 0.207558121 136 0.6749992 1661814000 0.791252962 0.208747038 137 0.6799992 1660157000 0.790464001 0.209535999 138 0.6849993 1657281000 0.789094628 0.210905372 139 0.6899993 1654847000 0.787935708 0.212064292 140 0.6949994 1651173000 0.786186377 0.213813623 141 0.6999994 1648863000 0.785086498 0.214913502 142 0.7049994 1647073000 0.784234210 0.215765790 143 0.7099964 1643798000 0.782674858 0.217325142 144 0.7149965 1642285000 0.781954461 0.218045539 145 0.7199965 1639299000 0.780532713 0.219467287 146 0.7249965 1637414000 0.779635193 0.220364807 147 0.7299966 1636452000 0.779177148 0.220822852 148 0.7349966 1628740000 0.775505171 0.224494829 149 0.7399966 1625014000 0.773731080 0.226268920 150 0.7449967 1620923000 0.771783199 0.228216801 151 0.7499967 1614048000 0.768509750 0.231490250 152 0.7549968 1572332000 0.748647173 0.251352827 153 0.7599968 1538880000 0.732719401 0.267280599 154 0.7649968 1509938000 0.718939012 0.281060988 155 0.7699969 1475289000 0.702441303 0.297558697 156 0.7749969 1455842000 0.693181845 0.306818155 157 0.7799969 1442941000 0.687039188 0.312960812 158 0.7849970 1431066000 0.681385048 0.318614952 159 0.7899970 1424536000 0.678275866 0.321724134 160 0.7949971 1425764000 0.678860563 0.321139437 161 0.7999971 1423645000 0.677851627 0.322148373

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162 0.8049971 1419990000 0.676111342 0.323888658 163 0.8099972 1418877000 0.675581400 0.324418600 164 0.8149973 1414926000 0.673700179 0.326299821 165 0.8199973 1407175000 0.670009632 0.329990368 166 0.8249973 1400646000 0.666900927 0.333099073 167 0.8299974 1398236000 0.665753434 0.334246566 168 0.8349974 1395787000 0.664587372 0.335412628 169 0.8399974 1391520000 0.662555690 0.337444310 170 0.8449975 1387382000 0.660585431 0.339414569 171 0.8499975 1384674000 0.659296049 0.340703951 172 0.8549976 1383281000 0.658632788 0.341367212 173 0.8599976 1379073000 0.656629199 0.343370801 174 0.8649976 1373542000 0.653995680 0.346004320 175 0.8699977 1368289000 0.651494526 0.348505474 176 0.8749977 1363791000 0.649352857 0.350647143 177 0.8799977 1356795000 0.646021795 0.353978205 178 0.8849978 1350951000 0.643239244 0.356760756 179 0.8899978 1347886000 0.641779880 0.358220120 180 0.8949979 1344600000 0.640215291 0.359784709 181 0.8999979 1339413000 0.637745562 0.362254438 182 0.9049979 1332283000 0.634350698 0.365649302 183 0.9099980 1328488000 0.632543754 0.367456246 184 0.9149981 1326108000 0.631410545 0.368589455 185 0.9199980 1322035000 0.629471234 0.370528766 186 0.9249981 1315913000 0.626556317 0.373443683 187 0.9299982 1309233000 0.623375714 0.376624286 188 0.9349982 1308340000 0.622950523 0.377049477 189 0.9399982 1305959000 0.621816838 0.378183162 190 0.9449983 1301708000 0.619792775 0.380207225 191 0.9499983 1298405000 0.618220091 0.381779909 192 0.9549984 1295889000 0.617022128 0.382977872 193 0.9599984 1293660000 0.615960816 0.384039184 194 0.9649984 1292124000 0.615229468 0.384770532 195 0.9699985 1290240000 0.614332423 0.385667577 196 0.9749985 1287723000 0.613133984 0.386866016 197 0.9799985 1286310000 0.612461201 0.387538799 198 0.9849986 1284448000 0.611574632 0.388425368 199 0.9899986 1282858000 0.610817572 0.389182428 200 0.9949987 1281236000 0.610045276 0.389954724 201 0.9999987 1279956000 0.609435819 0.390564181 202 1.0049990 1278588000 0.608784462 0.391215538 203 1.0099990 1276507000 0.607793619 0.392206381 204 1.0149990 1275173000 0.607158451 0.392841549 205 1.0199990 1275191000 0.607167021 0.392832979 206 1.0249990 1274751000 0.606957520 0.393042480 207 1.0299990 1273366000 0.606298069 0.393701931 208 1.0349990 1272299000 0.605790030 0.394209970 209 1.0399990 1270800000 0.605076299 0.394923701 210 1.0449990 1267591000 0.603548372 0.396451628 211 1.0499990 1261418000 0.600609171 0.399390829 212 1.0549990 1257053000 0.598530828 0.401469172 213 1.059996 1254794000 0.597455232 0.402544768 214 1.064996 1253222000 0.596706743 0.403293257 215 1.069996 1250611000 0.595463547 0.404536453

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216 1.074996 1248123000 0.594278915 0.405721085 217 1.079996 1247163000 0.593821822 0.406178178 218 1.084996 1246476000 0.593494716 0.406505284 219 1.089996 1246198000 0.593362349 0.406637651 220 1.094996 1243424000 0.592041542 0.407958458 221 1.099996 1241707000 0.591224013 0.408775987 222 1.104996 1237655000 0.589294701 0.410705299 223 1.109996 1235373000 0.588208154 0.411791846 224 1.114997 1232413000 0.586798785 0.413201215 225 1.119997 1229599000 0.585458933 0.414541067 226 1.124997 1225575000 0.583542953 0.416457047 227 1.129997 1215646000 0.578815378 0.421184622 228 1.134997 1209343000 0.575814279 0.424185721 229 1.139997 1205760000 0.574108277 0.425891723 230 1.144997 1201617000 0.572135637 0.427864363 231 1.149997 1197409000 0.570132047 0.429867953 232 1.154997 1192798000 0.567936575 0.432063425 233 1.159997 1187307000 0.565322100 0.434677900 234 1.164997 1181090000 0.562361950 0.437638050 235 1.169997 1175739000 0.559814135 0.440185865 236 1.174997 1172579000 0.558309538 0.441690462 237 1.179997 1169895000 0.557031584 0.442968416 238 1.184997 1165491000 0.554934671 0.445065329 239 1.189997 1159345000 0.552008327 0.447991673 240 1.194997 1153677000 0.549309576 0.450690424 241 1.199997 1143159000 0.544301555 0.455698445 242 1.204997 1136216000 0.540995729 0.459004271 243 1.209997 1128296000 0.537224715 0.462775285 244 1.214997 1121986000 0.534220283 0.465779717 245 1.219997 1116247000 0.531487727 0.468512273 246 1.224997 1109900000 0.528465678 0.471534322 247 1.229998 1104438000 0.525865012 0.474134988 248 1.234998 1100387000 0.523936177 0.476063823 249 1.239998 1095650000 0.521680710 0.478319290 250 1.244998 1091404000 0.519659028 0.480340972 251 1.249998 1085019000 0.516618886 0.483381114 252 1.254998 1082490000 0.515414733 0.484585267 253 1.259998 1077494000 0.513035947 0.486964053 254 1.264998 1072522000 0.510668588 0.489331412 255 1.269998 1066274000 0.507693678 0.492306322 256 1.274998 1061595000 0.505465827 0.494534173 257 1.279998 1055205000 0.502423305 0.497576695 258 1.284998 1049320000 0.499621232 0.500378768 259 1.289998 1042076000 0.496172088 0.503827912 260 1.294998 1035247000 0.492920541 0.507079459 261 1.299998 1029670000 0.490265118 0.509734882 262 1.304998 1023450000 0.487303539 0.512696461 263 1.309998 1017634000 0.484534320 0.515465680 264 1.314998 1012434000 0.482058402 0.517941598 265 1.319998 1007312000 0.479619623 0.520380377 266 1.324998 1004019000 0.478051700 0.521948300 267 1.329998 9979013000 0.475138830 0.524861170 268 1.334998 9951386000 0.473823403 0.526176597 269 1.339998 9905331000 0.471630549 0.528369451

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270 1.344998 9856695000 0.469314804 0.530685196 271 1.349998 9814702000 0.467315357 0.532684643 272 1.354998 9764512000 0.464925620 0.535074380 273 1.359998 9734268000 0.463485588 0.536514412 274 1.364999 9689147000 0.461337205 0.538662795 275 1.369999 9658373000 0.459871938 0.540128062 276 1.374999 9616029000 0.457855779 0.542144221 277 1.379999 9578287000 0.456058738 0.543941262 278 1.384999 9545581000 0.454501481 0.545498519 279 1.389999 9511935000 0.452899467 0.547100533 280 1.394999 9492741000 0.451985567 0.548014433 281 1.399999 9470316000 0.450917828 0.549082172 282 1.404996 9456678000 0.450268470 0.549731530 283 1.409996 9431958000 0.449091457 0.550908543 284 1.414996 9418806000 0.448465240 0.551534760 285 1.419996 9399645000 0.447552912 0.552447088 286 1.424996 9373379000 0.446302288 0.553697712 287 1.429996 9358088000 0.445574225 0.554425775 288 1.434996 9312887000 0.443422033 0.556577967 289 1.439996 9287639000 0.442219880 0.557780120 290 1.444996 9255890000 0.440708189 0.559291811 291 1.449996 9229478000 0.439450613 0.560549387 292 1.454996 9221811000 0.439085558 0.560914442 293 1.459996 9186116000 0.437385983 0.562614017 294 1.464996 9172956000 0.436759385 0.563240615 295 1.469996 9153393000 0.435827916 0.564172084 296 1.474996 9133137000 0.434863451 0.565136549 297 1.479996 9126587000 0.434551580 0.565448420 298 1.484996 9115246000 0.434011592 0.565988408 299 1.489996 9109546000 0.433740193 0.566259807 300 1.494996 9090516000 0.432834103 0.567165897 301 1.499997 9064375000 0.431589430 0.568410570 302 1.500003 9081825000 0.432420291 0.567579709

BRIDGE RAIL DESIGN PROCEDURE by EMAD BADIEE · PDF file · 2015-02-18BRIDGE RAIL...

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