Finite Element Analysis of Dapped-Ended Concrete Girders ...

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2020-01-30

Finite Element Analysis of Dapped-Ended Concrete

Girders Reinforced with Steel Headed Studs

Yuen, Kevin Wing-Chung

Yuen, K. W.-C. (2020). Finite Element Analysis of Dapped-Ended Concrete Girders Reinforced with

Steel Headed Studs (Unpublished master's thesis). University of Calgary, Calgary, AB.

http://hdl.handle.net/1880/111610

master thesis

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UNIVERSITY OF CALGARY

Finite Element Analysis of Dapped-Ended Concrete Girders Reinforced with Steel Headed Studs

by

Kevin Wing-Chung Yuen

A THESIS

SUBMITTED TO THE FACULTY OF GRADUATE STUDIES

IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE

DEGREE OF MASTER OF SCIENCE

GRADUATE PROGRAM IN CIVIL ENGINEERING

CALGARY, ALBERTA

JANUARY, 2020

© Kevin Wing-Chung Yuen 2020

i

Abstract

The application of dapped-ended girders in concrete construction arises in situations where it is

desired to maintain continuity between adjacent members. Dapping involves recessing the end of

a girder such that it can be placed on to supporting components. Due to the sudden decrease in

cross sectional area, significant shear strength in the member is lost and the introduction of a re-

entrant corner makes the girder prone to shear cracking. Careful shear reinforcement must

therefore be provided at the re-entrant corner.

Using headed studs in place of conventional reinforcement was first proposed by Herzinger and

El-Badry (2007) at the University of Calgary. They performed experiments which showed that

headed studs were effective in maintaining the strength and ductility of girders reinforced using a

combination of horizontal, vertical, and inclined configurations. The experimental data was

compared with the shear friction and diagonal bending methods of analysis. Overall, the

experiments and analytical methods showed good agreement, but the accuracy of the methods

varied in certain situations depending on the layout of the headed studs.

The current study is a finite element analysis (FEA) of seven of the specimens tested by

Herzinger and El-Badry (2007) using software ABAQUS. Results show that the diagonal

bending method is more suited for specimens with inclined reinforcement while shear friction

best predicts members with only horizontal and vertical reinforcement. As well, in analyzing

dapped-ended concrete girders, a variety of parameters can impact the results of the model, but

proper calibration can lead to the models’ suitability in being used for future parametric studies.

ii

Acknowledgements

I would first like to thank Dr. Mamdouh El-Badry for his guidance, advice, and patience

throughout my research. Your kindness and understanding have made this challenging endeavor

more surmountable. I would also like to express my deepest gratitude towards the Natural

Sciences and Engineering Research Council (NSERC) and Department of Civil Engineering for

their financial support. My gratitude also goes towards Dan Tilleman of the Civil Engineering

lab for procuring the relevant experimental components for my reference, as well as Doug

Phillips from the High Performance Computing department for his help and guidance in using

the university’s computing systems. Many thanks go towards Yadong Zhang, Qiang Chen, and

Bach Dinh Thang for their extremely valuable input in using ABAQUS and conversations about

best practices in the finite element method. As well, I would like to thank Chee Wong for giving

me a warm welcome to the graduate department during my first few days, and keeping me social

among my graduate peers within the department.

I would like to thank all my friends in our group known as “The Squad”. I have played, laughed,

and had countless memories with many of you throughout all of undergrad, and some of you

while in my grad studies. Your friendship, encouragement, and jokes have kept my motivation in

check for the past three years and beyond.

Finally, I would like to thank my parents and brother for their love, support, advice, and

encouragement in the completion of my graduate program. Without them, it would not have been

possible for me to reach this far in my journey.

iii

Table of Contents

Abstract ............................................................................................................................................ i

Acknowledgements ......................................................................................................................... ii

Table of Contents ........................................................................................................................... iii

List of Figures ................................................................................................................................. v

List of Tables .................................................................................................................................. x

List of Symbols .............................................................................................................................. xi

CHAPTER 1 - INTRODUCTION ............................................................................................ 1

1.1 General ............................................................................................................................. 1

1.2 Objectives and Scope ....................................................................................................... 3

1.3 Research Significance ...................................................................................................... 3

1.4 Thesis Organization.......................................................................................................... 4

CHAPTER 2 - BACKGROUND AND LITERATURE REVIEW .......................................... 5

2.1 Analytical Procedures for Dapped-ended Girder Design ................................................. 5

2.1.1 Design Using Strut-and-Tie Models ......................................................................... 5

2.1.2 Analysis Using Shear Friction .................................................................................. 8

2.1.3 Analysis Using Diagonal Bending ............................................................................ 9

2.2 Experimental Program by Herzinger and El-Badry (2007) ........................................... 11

2.3 Previous Work on Numerical Studies of Dapped-Ended Girders .................................. 19

2.3.1 Popescu et al. (2014)............................................................................................... 19

2.3.2 Moreno and Meli (2013) ......................................................................................... 21

2.3.3 Argirova et al. (2014) ............................................................................................. 23

2.4 Other Concrete Structures Modelled in ABAQUS ........................................................ 25

2.4.1 Genikomsou and Polak (2014)................................................................................ 25

2.4.2 Demir et al. (2017) .................................................................................................. 27

2.4.3 Zheng et al. (2010) .................................................................................................. 28

CHAPTER 3 - MODELLING APPROACH .......................................................................... 30

3.1 Specimen Representation ............................................................................................... 30

3.2 Meshing .......................................................................................................................... 31

3.3 Materials ......................................................................................................................... 38

iv

3.3.1 Concrete .................................................................................................................. 38

3.3.2 Steel......................................................................................................................... 51

3.4 Contact Interactions and Constraints ............................................................................. 54

3.5 Loads and Boundary Conditions .................................................................................... 61

3.6 Solution Technique and Analysis Steps ......................................................................... 64

CHAPTER 4 - RESULTS OF SENSITIVITY ANALYSES .................................................. 71

4.1 Sensitivity Analyses ....................................................................................................... 71

4.2 Results of the Sensitivity Analyses ................................................................................ 73

4.2.1 Reference Models ................................................................................................... 73

4.2.2 Remaining Models .................................................................................................. 86

CHAPTER 5 - RESULTS FROM ADDITIONAL STUDIES ............................................... 91

5.1 General ........................................................................................................................... 91

5.1.1 Reference Specimens .............................................................................................. 91

5.1.2 Remaining Specimens ........................................................................................... 117

5.1.3 Energy Check ........................................................................................................ 137

5.1.4 Effect of Stud Head Bearing Concrete.................................................................. 138

5.2 Summary of Primary Analyses .................................................................................... 142

5.3 Additional Refinements to Concrete Girder Mesh ....................................................... 143

CHAPTER 6 - DISCUSSION ............................................................................................... 148

6.1 General ......................................................................................................................... 148

6.2 Effect of Mesh Size ...................................................................................................... 148

6.3 Effect of Dilation Angle ............................................................................................... 149

6.4 Effect of Concrete Damage .......................................................................................... 152

6.5 Effect of Mesh Refinement around Cavities Created by Reinforcement ..................... 153

6.6 Implications for Shear Friction and Diagonal Bending Methods ................................ 154

CHAPTER 7 - SUMMARY, CONCLUSIONS, & FUTURE WORK ................................. 156

7.1 Summary ...................................................................................................................... 156

7.2 Conclusions .................................................................................................................. 156

7.3 Recommendations for Future Work ............................................................................. 158

References ................................................................................................................................... 159

Appendix I – E-mail Correspondence Regarding Permission to Use Content ........................... 162

v

List of Figures

Figure 1-1: Dapped-ended girder used in a bridge ........................................................................ 1

Figure 2-1: Typical B and D-regions of a dapped-ended girder.................................................... 6

Figure 2-2: Strut-and-tie models used for girder design by Herzinger and El-Badry (2007), with

(a) Model A, (b) Model B; and (c)Model C as combination of Models A and B. Dashed lines

represent compressive struts while solid lines symbolize tension ties ............................................ 7

Figure 2-3: Free body diagram of cracked section based on shear friction. Adapted from

Herzinger and El-Badry (2007) ...................................................................................................... 8

Figure 2-4: Free body diagram of cracked section using diagonal bending and resulting strain

distribution. Adapted from Herzinger and El-Badry (2007) ........................................................ 10

Figure 2-5: Experimental setup and strain gauge placement for shear displacement

measurement (inset). From Herzinger and El-Badry (2007)........................................................ 13

Figure 2-6: Specimens tested by Herzinger and El-Badry (2007) (a) DE-A-0.5/1.0 (b) DE-B-

0.5/1.0 (c) DE-C-1.0 (d) DE-D-1.0 (e) DE-D*-1.0 ...................................................................... 16

Figure 2-7: Predicted vs. actual crack patterns (Herzinger and El-Badry 2007) (a) DE-A-0.5/1.0

(b) DE-B-0.5/1.0 (c) DE-C-1.0 (d) DE-D-1.0; and (e) DE-D*-1.0 .............................................. 17

Figure 2-8: Specimen DE-D*-1.0 reinforcement strains (Herzinger and El-Badry 2007) (a)

Horizontal stud strains (b) Stirrup and vertical stud strains (c) Inclined stud strains (d) Load-45°

displacement curves for all specimens .......................................................................................... 18

Figure 3-1: Overview of typical model developed in ABAQUS ................................................... 31

Figure 3-2: 3D geometries (left) and meshes (right) of (a) hinge assembly; and (b) roller

assembly ........................................................................................................................................ 33

Figure 3-3: Spreader beam 3D geometry (left) and mesh (right) ................................................ 34

Figure 3-4: 3D view reinforcing cages (left) and their meshes (right) of Specimens: (a) DE-B-

0.5/1.;0 (b) DE-C-1.0; and (c) DE-D-1.0 ..................................................................................... 35

Figure 3-5: Cut view of cavity introduced into the girder of DE-B-0.5/1.0 by the ¾” single-

headed stud ................................................................................................................................... 37

Figure 3-6: Girder partition geometries (left) and typical meshes (right) of Specimens: (a) DE-B-

0.5/1.0 (b) DE-C-1.0; and (c) DE-D-1.0 ...................................................................................... 38

Figure 3-7: Yield surface viewed from biaxial principal stress plane. Adapted from ABAQUS

User’s Manual (Similia 2013) ...................................................................................................... 42

Figure 3-8: Yield surface viewed from deviatoric stress plane, with variations in the ratio Kc.

Adapted from ABAQUS User’s Manual (Similia 2013) ............................................................... 42

Figure 3-9: Compressive stress-strain curve. Adapted from Wahalathantri et al. (2011) ........... 45

Figure 3-10: Tension-displacement curve for concrete. Adopted from CEB-FIP (2010) ............ 47

Figure 3-11: Definition of inelastic and plastic strains in compression (Similia 2013) .............. 49

Figure 3-12: Actual (left) vs. adopted (right) stress strain curve for mild and headed stud steels

....................................................................................................................................................... 52

vi

Figure 3-13: Interacting surfaces, highlighted in red, specified between the inner concrete

surfaces created by the studs (top) and the studs themselves (bottom). Specimen DE-D-1.0 shown

....................................................................................................................................................... 55

Figure 3-14: Typical bond-slip law for reinforcement in pullout (left) and traction separation

model available in ABAQUS (right) ............................................................................................. 57

Figure 3-15: Interacting surfaces highlighted in red, for the roller (left) and hinge (right)

support assemblies ........................................................................................................................ 58

Figure 3-16: Interacting surfaces, highlighted in red, between spreader beam support roller and

girder load bearing plate. The roller cylinder is truncated at the connection to the spreader

beam to enable a definite, flat surface to be present in order for the roller to be tied to the

bearing plate above. ABAQUS cannot tie surfaces which are tangent to one another. ............... 59

Figure 3-17: Thin strips of area belonging to the girder overlapped by the support faces were

ignored when applying the tie constraints, shown by the yellow arrows ..................................... 60

Figure 3-18: The sections of the girders in red were treated as separate parts and tied to the

main girder body, shown by the highlights: (a) DE-B-0.5/1.0; (b) DE-C-1.0; and (c) DE-D-1.0 61

Figure 3-19: z-symmetry boundary condition applied to the central plane of girders ................ 63

Figure 3-20: Zero z-displacement boundary condition applied to 3/8” half stud ........................ 63

Figure 4-1: Specimen DE-A-0.5 mesh analysis (full damage, ψ=35°) ........................................ 76

Figure 4-2: Specimen DE-A-0.5 damage regime analysis (mesh size=15 mm, ψ=35°) .............. 76

Figure 4-3: Specimen DE-A-0.5 dilation angle analysis (mesh size=15 mm, tension damage only)

....................................................................................................................................................... 77

Figure 4-4: Specimen DE-B-0.5 mesh analysis (full damage, ψ=35°) ........................................ 78

Figure 4-5: Specimen DE-B-0.5 damage regime analysis (mesh size=15 mm, ψ=35°) .............. 79

Figure 4-6: Specimen DE-B-0.5 dilation angle analysis (mesh size=15 mm, tension damage only)

....................................................................................................................................................... 79

Figure 4-7: Specimen DE-C-1.0 mesh analysis (full damage, ψ=35°) ........................................ 82

Figure 4-8: Specimen DE-C-1.0 damage regime analysis (mesh size=15 mm, ψ=35°) .............. 82

Figure 4-9: Specimen DE-C-1.0 dilation angle analysis (mesh size=15 mm, tension damage only)

....................................................................................................................................................... 83

Figure 4-10: Specimen DE-D-1.0 mesh analysis (full damage, ψ=35°) ...................................... 84

Figure 4-11: Specimen DE-D-1.0 damage regime analysis (mesh size=15 mm, ψ=35°) ............ 85

Figure 4-12: The applied displacement for Specimen DE-D-1.0 was increased from 25 mm to 40

mm in order for failure to occur (mesh size=15 mm, ψ=35°, tension damage only) ................... 85

Figure 4-13: Specimen DE-A-1.0 results (mesh size=15 mm, ψ=50°, tension damage only) ........ 88

Figure 4-14: Specimen DE-B-1.0 sensitivity results (mesh size=15 mm, tension damage only, ψ=55°)

....................................................................................................................................................... 89

Figure 4-15: Specimen DE-D*-1.0 results (mesh size=15 mm, tension damage only, ψ=35°) ...... 90

Figure 5-1: Effect of concrete Poisson’s ratio on Specimen DE-A-0.5 (mesh size=15 mm, tension

damage only, ψ=50°) .................................................................................................................... 92

Figure 5-2: Specimen DE-A-0.5 stirrup strains at east end ......................................................... 93

vii

Figure 5-3: Specimen DE-A-0.5 stirrup strains at west end ........................................................ 93

Figure 5-4: Specimen DE-A-0.5 20M horizontal reinforcement strains ...................................... 94

Figure 5-5: Specimen DE-A-0.5 average flexural reinforcement strains ..................................... 94

Figure 5-6: Specimen DE-A-0.5 vertical deflection at ends and quarter points of full depth at

different load levels. Dotted lines show experimental values ....................................................... 95

Figure 5-7: Specimen DE-A-0.5 failure patterns at (a) peak load; (b) numerical failure; (c)

numerical failure corresponding to experimental shear displacement at failure; and (d)

experimental failure ...................................................................................................................... 96

Figure 5-8: Effect of concrete Poisson’s ratio on Specimen DE-B-0.5 (mesh size=15 mm, tension

damage only, ψ=55°) .................................................................................................................... 98

Figure 5-9: Effect of mesh size of concrete surrounding vertical stud heads on Specimen DE-B-

0.5 (mesh size=15 mm, tension damage only, ψ=55°, ν=0) ......................................................... 99

Figure 5-10: Specimen DE-B-0.5 vertical stud strains ................................................................ 99

Figure 5-11: Specimen DE-B-0.5 horizontal stud strains .......................................................... 100

Figure 5-12: Specimen DE-B-0.5 stirrup strains ....................................................................... 100

Figure 5-13: Specimen DE-B-0.5 average flexural reinforcement strains ................................. 101

Figure 5-14: Specimen DE-B-0.5 vertical deflection at ends and quarter points of full depth at

different load levels. Dotted lines show experimental values ..................................................... 101

Figure 5-15: Specimen DE-B-0.5 failure patterns at (a) peak load; (b) numerical failure; (c)


experimental failure .................................................................................................................... 103

Figure 5-16: Effect of concrete Poisson’s ratio on Specimen DE-C-1.0 (mesh size=15 mm,

tension damage only, ψ=30°) ..................................................................................................... 105

Figure 5-17: Specimen DE-C-1.0 stirrup strains at east end ..................................................... 106

Figure 5-18: Specimen DE-C-1.0 stirrup strains at west end .................................................... 106

Figure 5-19: Specimen DE-C-1.0 U-stirrup strains ................................................................... 107

Figure 5-20: Specimen DE-C-1.0 inclined stud strains.............................................................. 107

Figure 5-21: Specimen DE-C-1.0 average flexural reinforcement strains ................................ 108

Figure 5-22: Specimen DE-C-1.0 vertical deflection at ends and quarter points of full depth at

different load level. Dotted lines show experimental values ....................................................... 108

Figure 5-23: Specimen DE-C-1.0 failure patterns at (a) peak load; (b) numerical failure; (c)



Figure 5-24: Effect of concrete Poisson’s ratio on Specimen DE-D-1.0 (mesh size=15 mm,


Figure 5-25: Specimen DE-D-1.0 vertical stud strains .............................................................. 111

Figure 5-26: Specimen DE-D-1.0 inclined stud strains ............................................................. 112

Figure 5-27: Specimen DE-D-1.0 horizontal stud strains .......................................................... 112

Figure 5-28: Specimen DE-D-1.0 stirrup strains at east end ..................................................... 113

Figure 5-29: Specimen DE-D-1.0 stirrup strains at west end .................................................... 113

viii

Figure 5-30: Specimen DE-D-1.0 average flexural reinforcement strains ................................ 114

Figure 5-31: Specimen DE-D-1.0 vertical deflection at ends and quarter points of full depth at


Figure 5-32: Specimen DE-D-1.0 failure patterns at (a) peak load; (b) numerical failure; (c)



Figure 5-33: Effect of concrete Poisson’s ratio on Specimen DE-A-1.0 (mesh size=15 mm,


Figure 5-34: Specimen DE-A-1.0 stirrup strains at east end ..................................................... 119

Figure 5-35: Specimen DE-A-1.0 stirrup strains at west end .................................................... 119

Figure 5-36: Specimen DE-A-1.0 20M horizontal reinforcement strains .................................. 120

Figure 5-37: Specimen DE-A-1.0 average flexural reinforcement strains ................................. 120



Figure 5-39: Specimen DE-A-1.0 failure patterns at (a) peak load; (b) numerical failure, and

numerical failure corresponding to experimental shear displacement at failure; and (c)


Figure 5-40: Effect of concrete Poisson’s ratio on Specimen DE-B-1.0 (mesh size=15 mm,


Figure 5-41: Specimen DE-B-1.0 vertical stud strains .............................................................. 124

Figure 5-42: Specimen DE-B-1.0 horizontal stud strains .......................................................... 125

Figure 5-43: Specimen DE-B-1.0 stirrup strains ....................................................................... 125

Figure 5-44: Specimen DE-B-1.0 average flexural reinforcement strains ................................. 126


different load level. Dotted lines show experimental results ...................................................... 126

Figure 5-46: Specimen DE-B-1.0 failure patterns at (a) peak load; (b) numerical failure; (c)



Figure 5-47: Effect of concrete Poisson’s ratio on Specimen DE-D*-1.0 (mesh size=15 mm,


Figure 5-48 Effect of dilation angle on Specimen DE-D*-1.0 (tension damage only, mesh

size=15 mm, ν=0) ....................................................................................................................... 131

Figure 5-49: Specimen DE-D*-1.0 vertical stud strains ............................................................ 131

Figure 5-50: Specimen DE-D*-1.0 inclined stud strains ........................................................... 132

Figure 5-51: Specimen DE-D*-1.0 horizontal stud strains ........................................................ 132

Figure 5-52: Specimen DE-D*-1.0 stirrup strains at east end ................................................... 133

Figure 5-53: Specimen DE-D*-1.0 stirrup strains at west end .................................................. 133

Figure 5-54: Specimen DE-D*-1.0 average flexural reinforcement strains .............................. 134

Figure 5-55: Specimen DE-D*-1.0 vertical deflection at ends and quarter points of full depth at

different load level. Dotted lines show experimental results ...................................................... 134

ix

Figure 5-56: Specimen DE-D*-1.0 failure patterns at (a) peak load; (b) numerical failure; (c)



Figure 5-57: Kinetic to internal energy ratio for all specimens, taken from the best cases in the

additional (final) studies (mesh size=15 mm, tension damage only, ν=0.2). Values beyond 12%

are not shown. ............................................................................................................................. 138

Figure 5-58: Concrete bearing stress adjacent to horizontal studs in Specimen DE-B-0.5. Mesh

size=15 mm, dilation angle=55°, tension damage only, concrete Poisson’s ratio=0................ 139

Figure 5-59: Concrete bearing stress adjacent to vertical studs in Specimen DE-B-0.5. Mesh


Figure 5-60: Concrete bearing stress adjacent to inclined studs in Specimen DE-C-1.0. Mesh


Figure 5-61: Concrete bearing stress adjacent to horizontal studs in Specimen DE-D-1.0. Mesh


Figure 5-62: Concrete bearing stress adjacent to inclined studs in Specimen DE-D-1.0. Mesh




Figure 5-64: Effect of mesh refinement on the behaviour of Specimen DE-B-0.5. Tension damage

only, concrete Poisson’s ratio=0. ............................................................................................... 144

Figure 5-65: Effect of mesh refinement on the behaviour of Specimen DE-D-1.0. Tension

damage only, concrete Poisson’s ratio=0. ................................................................................. 146

Figure 5-66: Effect of increased applied displacement from 40 mm to 60 mm on the behaviour of

Specimen DE-D-1.0. Tension damage only, concrete Poisson’s ratio=0. ................................. 146

Figure 6-1: “Snap-back” behaviour present in east horizontal stud of Specimen DE-B-1.0.

Results for trial simulation shown, not used for sensitivity analyses. ........................................ 150

Figure 6-2: Possible source of compression damage at crack location ..................................... 153

x

List of Tables

Table 2-1: Summary of results by Herzinger and El-Badry (2007).............................................. 18

Table 3-1: Concrete properties (Herzinger 2007) ........................................................................ 51

Table 3-2: Steel properties (Herzinger 2007) ............................................................................... 53

Table 4-1: Summary of results of sensitivity analyses of Specimen DE-A-0.5 ............................. 75

Table 4-2: Summary of results of sensitivity analyses of Specimen DE-B-0.5 ............................. 78

Table 4-3: Summary of results of sensitivity analyses of Specimen DE-C-1.0 ............................. 81

Table 4-4: Summary of results of sensitivity analyses of Specimen DE-D-1.0 ............................. 86

Table 4-5: Summary of results of sensitivity analyses of Specimen DE-A-1.0 ............................. 87

Table 4-6: Summary of results of sensitivity analyses of Specimen DE-B-1.0 ............................. 89

Table 4-7: Summary of results of sensitivity analyses of Specimen DE-D*-1.0 ........................... 90

Table 5-1: Summary of Specimen DE-A-0.5 results from Poisson’s ratio study .......................... 92

Table 5-2: Summary of Specimen DE-B-0.5 results from additional studies ............................... 98

Table 5-3: Summary of Specimen DE-C-1.0 results from additional studies ............................. 104

Table 5-4: Summary of Specimen DE-D-1.0 results from additional studies............................. 110

Table 5-5: Summary of Specimen DE-A-1.0 results from additional studies ............................. 118

Table 5-6: Summary of Specimen DE-B-1.0 results from additional studies ............................. 123

Table 5-7: Summary of Specimen DE-D*-1.0 results from additional studies........................... 130

Table 5-8: Summary of specimen results for final cases selected .............................................. 142

Table 5-9: Summary of Specimen DE-B-0.5 results from mesh refinement ............................... 143

Table 5-10: Summary of Specimen DE-D-1.0 results from mesh refinement ............................. 145

xi

List of Symbols

Chapter 2

As = area of steel

b = beam width

c = neutral axis depth

df = parallel-to-crack distance from flexural rebar cut to neutral axis

di = parallel-to-crack distance from diagonal rebar cut to neutral axis

ds1 = parallel-to-crack distance from stirrup cut to neutral axis

f’c = concrete compressive strength

fs = steel stress

fs,longitudinal = flexural rebar stress

fs,inclined = diagonal rebar stress

fs,vertical = stirrup stress

fy = steel yield stress

E = Young’s modulus of general material

Es = Young’s modulus of steel

F = force in flexural rebar

Fi = force in diagonal rebar

h = beam depth

H = horizontal reaction at beam support

ld = development length

ld,provided = provided length

P = applied load

R = normal force at crack interface

S = frictional force at crack interface

V = vertical reaction at beam support

Vs1 = force in stirrup

V45 = shear resistance of concrete along a 45° plane

x = distance from bearing plate edge to reaction location

y = depth of rebar

α = diagonal reinforcement angle to horizontal

α1 = ratio of average compressive stress to strength

β1 = ratio of stress block depth to neutral axis depth

βv = adjustment factor relating shear resistance to compressive strength and member depth

εcu = concrete strain

εy = steel strain at y

λ = concrete density-strength factor

ψ = flexural curvature

ϕc = concrete resistance factor

ϕs = steel resistance factor

θ = crack angle to horizontal

xii

Chapter 3

a = expression for smooth step polynomial

Ai = polynomial amplitude at time ti

B = strain state matrix related to shape function matrix N

cd = material dilational wave speed

C = damping matrix

d, di = damage variable, may be tensile or compressive (subscript i)

D = constitutive matrix

𝐃0𝑒𝑙 = initial elastic constitutive tensor

fb0 = initial equibiaxial compressive stress

fc = maximum uniaxial compressive stress

f’c, fck = concrete compressive strength

fc0, σc0 = initial uniaxial compressive

fcm = concrete mean compressive strength

fctm = concrete tensile strength

ft0, σt0 = initial uniaxial tensile yield stress

fu = steel ultimate stress

fy = steel yield stress

Eh = steel hardening modulus

Es = Young’s modulus for steel

E0 = initial Young’s modulus of concrete

F = yield/failure criterion expression

Fin = n

th entry of force vector at increment i

G = plastic potential function

GF = fracture energy parameter

I1 = first stress invariant of Cauchy stress tensor

I = identity tensor

Jinp

= entry of Jacobian matrix at nth

row and pth

column

J2 = second deviatoric stress invariant of Cauchy stress tensor

J3 = third deviatoric stress invariant of Cauchy stress tensor

Kc = stress invariant ratio

Knn, Kss, Ktt = bond-slip stiffness in normal, longitudinal, and tangential directions, respectively

K, K(e)

= overall/element stiffness matrix

l0 = characteristic specimen length

Le = characteristic element length

M = mass matrix

N = shape function matrix

�̅� = hydrostatic stress

p, p(e)

= overall/element vector of applied forces

pb, pb(e)

= overall/element vector of loads due to body forces

pi, pi(e)

= overall/element vector of loads due to initial deformations

ps, ps(e)

= overall/element vector of loads due to surface tractions

�̅� = Mises equivalent stress

s1 = relative interface slip up to τmax

s2, s3 = relative interface slip at stress decrease and failure, respectively

xiii

S1, S2, S3 = deviatoric stress directions

�̅� = effective stress deviator

t, ti = time variable; point in time at increment i

uim = m

th entry of the displacement vector at increment i

𝑢𝑡𝑐𝑘 = user defined cracking displacement in tension

𝑢𝑡𝑝𝑙

= user defined plastic displacement in tension

u = overall vector of nodal displacements

�̇�, �̇�𝑖 = velocity vector (at increment i)

�̈�, �̈�𝑖 = acceleration vector (at increment i)

w, w1, wc = crack displacements

α, γ = constants of concrete damaged plasticity yield criterion

β = constant of concrete damaged plasticity yield criterion; constant of concrete stress strain

curve

δ = plastic degradation variable

𝛥𝑢𝑖+1𝑚 = change in m

th entry of the displacement vector at increment i

Δt , Δt(i)

= incremental time step

ϵ = eccentricity of Ducker-Prager hyperbolic surface

εc = concrete compressive strain

εi = concrete strain, may be compressive or tensile

εnom = nominal strain

εpl = plastic strain (for steel)

εtru = true strain

εu = steel strain at ultimate stress

εy = steel yield strain

ε0 = concrete strain at peak stress

𝜀�̃�𝑖𝑛 = inelastic crushing strain

𝜀�̃�𝑝𝑙

= plastic strain, may be compressive or tensile

𝜀�̃�𝑝𝑙

= inelastic strain, may be compressive or tensile

𝜀�̃�𝑐𝑘 = cracking strain

ε = total strain tensor

εp, ε

pl = plastic strain tensor

ε0 = initial strain vector

�̇�𝑝 = flow rule expression

�̇� = plastic multiplier

ϕ = body force vector

Φ = surface traction vector

ψ = dilation angle of concrete

ρ = ratio of the square root of the second stress invariants in the tensile and compressive

meridians; density of general material

σc = concrete compressive stress strain function

σi = concrete stress, may be compressive or tensile

σnom = nominal stress

σmax = max principal stress

σt = concrete tensile stress strain function

σtru = true stress

xiv

σ1, σ2, σ3 = principal stress directions

c = effective cohesive stress in compression

𝜎𝑖 = effective cohesive stress, may be compressive or tensile

t = effective cohesive stress in tension

max = effective stress equivalent

�̅� = effective stress tensor

τf = failing shear stress for bond-slip interaction

τmax = maximum shear stress for bond-slip interaction

Chapter 4 and 5

ψ = Dilation angle of concrete

ν = Poisson’s ratio

1

CHAPTER 1 - INTRODUCTION

1.1 General

Dapped-ended girders are beam elements whose depth is reduced at their ends such that the

resulting “L”-shaped profile allows the girder to be placed onto adjacent supports or adjacent

girders without occupying additional space and creating a discontinuous appearance. Typically,

dapped-ended girders are supported at their ends either by column extrusions known as corbels,

or cantilevered members whose ends are also dapped – but inverted – to support the beam or

girder. Dapping is useful in structures such as bridges because it allows for the accommodation

of expansion joints, which are vital to ensure that traffic and environmental loads do not damage

the members. Figure 1-1 illustrates dapped-ended construction used at a bridge joint.

Figure 1-1: Dapped-ended girder used in a bridge

The proper design of dapped-ended girders is crucial because the re-entrant corners are

conducive to the formation of stress concentrations. The reduction in member depth implies a

reduction in effective depth available to resist shear forces. As a result, shear failure is likely to

occur assuming that sufficient longitudinal reinforcement is provided to resist flexure elsewhere.

2

This failure type is characterized by diagonal cracks extending upwards from the re-entrant

corner, which corresponds to the direction perpendicular to that of the maximum principal tensile

stress trajectories. Consequently, dapped-ended girders are usually heavily reinforced near the

reentrant corners. Congestion, however, is often a problem with conventional reinforcement

since it becomes more difficult during construction to cast around the bars. Additionally,

reinforcement spaced too close together may cause surrounding concrete to suffer from increased

cracking due to shrinkage, and the bond quality of the bars with the concrete may be reduced.

Several novel methods, however, have been developed to alleviate the issue of congestion.

Externally bonded fibre-reinforced sheets and pre-stressing tendons have been proposed to aid in

shear resistance.

The use of headed studs as shear reinforcement was proposed by Herzinger and El-Badry (2007)

to be used in place of conventional reinforcement, and has been shown to reduce congestion,

eliminating the need for external anchorage using angles or plates, and retaining the strength of

the member as well as its ductility. The authors performed the design of dapped-ended

specimens reinforced with combinations of horizontal, vertical, and inclined studs using strut-

and-tie models, used the shear friction and diagonal bending analysis methods to determine the

ultimate strengths and cracking patterns, and then proceeded to test their designs experimentally.

Their experimental data consist of load-strain and load-deflection curves, crack widths, and

ultimate loads. Agreement between the analytical and experimental methods was generally

achieved. The shear friction method was efficient in strength prediction of some specimens and

the diagonal bending method proved efficient in some others.

3

1.2 Objectives and Scope

The objective of the research presented herein is to conduct a numerical investigation, using the

commercial finite element (FEM) software ABAQUS, into the behaviour of the dapped-ended

girder specimens reinforced with different layouts of single and double headed studs tested by

Herzinger and El-Badry (2007). The purpose of the numerical investigation is to verify the

experimental results, providing insight into the relative accuracy of the analytical methods

developed for designing such girders. Comparison of the numerical results will be made with the

experimental data of the load versus shear deflections, cracking behaviour, stresses and strains in

the reinforcements, and failure modes. As well, the effect due to bearing of the stud heads on the

concrete will also be examined. Due to limitations with the FE software used, crack widths were

not included in the study. Sensitivity analyses will also be performed by varying mesh sizes and

certain material parameters for the specimens, and the validity of the results will be checked

from a modelling perspective. Where applicable, background and discussions leading up to the

development of the modelling characteristics will be provided as necessary.

1.3 Research Significance

The purpose of developing the numerical models in the current study was to validate the

effectiveness of using headed studs in place of conventional reinforcement in dapped-ended

concrete girders, as shown by the original experiments conducted. Together with the

experiments, results from the finite element analysis showed that headed studs are a viable

alternative to using stirrups and external anchoring components. The strengths and shear ductility

of specimens reinforced with headed studs is not compromised. Development of the benchmark

models will facilitate parametric studies to be performed in which different girder geometries,

stud layouts, and loading conditions can be investigated entirely within a numerical environment.

4

This will eliminate the necessity of redeveloping new experimental programs, which has positive

implications with regards to time and monetary costs.

1.4 Thesis Organization

The thesis is divided into seven chapters. Chapter 2 covers basic design procedures commonly

used for designing dapped-ended girders, and will review the experimental program of Herzinger

and El-Badry. Work performed by other researchers on the numerical modelling of dapped-

ended concrete girders will also be reviewed, as well as existing studies performed on concrete

structures using ABAQUS. Chapter 3 will cover the material properties used for the concrete

girders, conventional reinforcement, and headed studs, and will provide a detailed coverage of

modelling attributes within ABAQUS, including loads, boundary conditions, meshes, contact

interactions, and other techniques. Chapter 4 will provide the numerical results of the initial

studies according to the reference sensitivity parameters investigated; namely mesh size,

concrete dilation angle, and damage regime. In Chapter 5, the finite element results due to

refinements on the models from the previous chapter are presented, whereby simulation issues

are addressed and eliminated. In particular, the concrete Poisson’s ratio is adjusted for all

specimens for better ease of convergence. Chapter 6 is a discussion of the behaviour of the

dapped-ended girders provided by the finite element simulations, citing notable occurences

common to the specimens during analysis. Lastly, Chapter 7 will conclude the thesis and

summarize the overall procedures, numerical findings and observations, and provide

recommendations for future work.

5

CHAPTER 2 - BACKGROUND AND LITERATURE REVIEW

2.1 Analytical Procedures for Dapped-ended Girder Design

Traditional methods used for shear and flexural design of regular prismatic concrete members

are not applicable for members with discontinuities, at least, not within the vicinity of said

regions. Reinforcement proportioning and placement, and determination of strength capacities

were performed by Herzinger and El-Badry (2007) using three methods in the design of the

dapped-ended girders.

2.1.1 Design Using Strut-and-Tie Models

The strut-and-tie model, introduced by Schlaich et al. (1987), simplifies the analysis of a

member with complex geometry by treating it as a truss, with struts (concrete) acting in

compression (c) and ties (reinforcement) acting in tension (T). Struts and ties converge at

locations called nodes, which can be classified as CCC, CCT, CTT, or TTT, depending on the

number of struts and ties converging at that node. Locations with concentrated effects such as

applied loads and supports must always have nodes. The method considers St. Venant’s

principle, which states that local effects resulting from two different but statically equivalent

loads becomes insignificant at large distances from those effects. The model is also based on the

lower bound plasticity theorem, where, for some stress state experienced by the structure, static

equilibrium is satisfied internally and externally at all locations, and the equivalent stress remains

below yielding conditions. The application of this principle ensures conservatism. Designs must

always be such that ties will yield before the struts fail. Indeed, discontinuities contribute to

member complexity since Bernoulli’s beam theory no longer applies at these locations; in other

words, plane sections do not remain plane, normal stresses in the direction of the member

thickness are significant, and the neutral axis is not perpendicular to plane sections since shear

6

strain becomes relatively large (Owatsiriwong 2013). To this end, such members are usually

divided into regions to determine which design methods and behaviour are applicable. Disturbed,

or “D” regions are locations where elementary beam theory is invalid and where the strut-and-tie

method is applied. These regions need not be exclusively from geometry, but can also be at

locations where concentrated loads are applied. In contrast, “B” regions are treated to satisfy the

Bernoulli hypothesis; stresses in these regions may be determined from internal shear, bending,

torsional, or axial forces (1987). Figure 2-1 shows the “B” and “D” regions of a symmetric

dapped-ended girder loaded at four points.

Figure 2-1: Typical B and D-regions of a dapped-ended girder

Design using the strut-and-tie method is iterative since the relative angles of the struts and ties

determine the forces experienced by them, and hence the amount of reinforcement required.

Furthermore, nodal sizes may vary with different truss schemes which will consequently affect

the amount of anchorage needed. The reinforcement placement in the design of the specimens by

Herzinger and El-Badry is shown below in Figure 2-2. Two models, (A) and (B), were

considered separately and combined into a third model (C) for maximum effectiveness. From the

strut-and-tie schemes developed, it was clear that a combination of horizontal, vertical, and

diagonal reinforcement would provide the greatest advantage. The effective compressive

strength is highest if nodes only carry struts (CC(C); for nodes carrying ties the allowable stress

7

limit is only between 60 to 80 percent of compressive strength. Therefore, the use of headed

studs was beneficial since the bearing action from the heads would induce compression on the

nodes.

Figure 2-2: Strut-and-tie models used for girder design by Herzinger and El-Badry (2007), with

(a) Model A, (b) Model B; and (c)Model C as combination of Models A and B.

Dashed lines represent compressive struts while solid lines symbolize tension ties

It is important to note that “partial depth” ties such as DE will not in reality be designed to cut

off part way along the depth of the section.

(a) Model A (b) Model B

(c) Model C

8

2.1.2 Analysis Using Shear Friction

In this method, shear is assumed to be the governing failure mode, characterized by a diagonal

shear crack which will induce reactional forces in the transverse and longitudinal reinforcement.

Additionally, frictional forces will be introduced along the boundary of the cracked faces. As

always, equilibrium must be satisfied as a result of all internal forces, applied loads, and

reactions, as shown in the free body diagram of Figure 2-3 below.

Figure 2-3: Free body diagram of cracked section based on shear friction. Adapted from

Herzinger and El-Badry (2007)

The frictional force S and related normal force R may be expressed in terms of the reinforcement

forces and reactions. Therefore, the resulting static equation is given by (Herzinger and El-Badry

2007) as:

2 2

45 1

45

cos2 cot cot 1 cot cos cot sini

i s i

F F HV V F F H V F

V

(2-1)

which is quadratic in V since the horizontal reaction H is simply a fraction of V. In the above, V45

is the shear resistance of concrete in a rectangular section along a 45° plane, and thus given by:

'

45 c v cV f bh (2-2)

9

where λ is a density factor, ϕc is the material resistance factor for concrete, b and h are the

section depth and width respectively, and βv is an adjustment factor given empirically as

(Herzinger and El-Badry 2007)

0.25 0.25

'

30 500 0.36v

c

MPa mm

f h

(2-3)

which relates shear resistance to compressive strength and member depth. The longitudinal,

inclined, and vertical steel forces F, Fi, and 1sV , respectively, are given by

s s sA f (ϕs is the

steel material resistance factor and As is area of steel), which is either governed by the yield

stress fy or bond and anchorage, defined by the development length ld. In the latter, the stress is

provided as (Herzinger and El-Badry 2007)

,d provided

s y

d

lf f

l (2-4)

which must be less than fy. Since multiple crack angles are possible, the solution above should be

iterated for θ to produce the lowest possible V. Hence, from simple statics, the largest allowable

value for P/2 could be determined.

2.1.3 Analysis Using Diagonal Bending

In the diagonal bending method, flexural effects are taken into account when deriving the

equilibrium equation. This technique is therefore more accurate than shear friction when applied

to dapped-ended girders. Using this method, a shear crack forms in a similar manner compared

with shear friction, but terminates after propagating up to a certain depth. A portion of the

concrete depth remains under compression and the neutral axis is located at depth c of the

compression zone. As before, the reinforcement provides internal axial forces which must

10

balance out with all external loads, support reactions, and shear and normal forces generated in

the concrete in compression. This is illustrated in Figure 2-4 below.

Figure 2-4: Free body diagram of cracked section using diagonal bending and resulting strain

distribution. Adapted from Herzinger and El-Badry (2007)

If moments of all forces are taken about the crack tip, the following expression is obtained for

the vertical reaction (Herzinger and El-Badry 2007):

11 11 sin sin cos

2

cot

f i i s sRc H h c Fd Fd V d

Vx h c

(2-5)

where, from equilibrium of the horizontal forces:

cosiR H F F (2-6)

with R being the compressive force above the neutral axis defined by CSA A23.3 as:

'

1 1 c cR f bc (2-7)

where α1 is the ratio of average compressive stress to strength, β1 is the ratio of stress block depth

to neutral axis depth, and ϕc is the material resistance factor for concrete. As depicted, a linear

strain distribution is assumed with compatibility between strains above the neutral axis and in the

plane of cracking.

11

As a result, strains at any depth in the compression zone and along the direction of the crack

below the neutral axis can be related through similar triangles, with the strain at a depth y given

by:

sin

y cu

y c

c

(2-8)

where the maximum strain in concrete is εcu = 0.0035. With this considered, the resulting

longitudinal, inclined, and vertical reinforcement forces are then given, respectively, by

(Herzinger and El-Badry 2007):

, 2sin

f cu s

s longitudinal

d Ef

c

(2-9)

, 2sin

i cu ss inclined

d Ef

c

(2-10)

1, 2cos

s cu ss vertical

d Ef

c

(2-11)

with Es being the modulus of elasticity of steel. As with shear friction, the governing stress for

steel depends on its development length and yield strength. From Eqsuations (2-5) and (2-6), c

and V can be solved for a selected value of angle θ. The angle θ which minimizes Equation (2-5)

will define the failure plane inclination, whose value must again be iterated as before.

2.2 Experimental Program by Herzinger and El-Badry (2007)

Eleven dapped-ended girder specimens were fabricated for testing following their design and

analysis. The seven most critical specimens were chosen for the current study. For all specimens,

the supports were a basic hinge-roller combination, configured such that a horizontal reaction

equal to 20 percent of the vertical reaction would be induced at both ends. The presence of

horizontal axial loads was important in simulating the effects of creep and shrinkage, which

12

would induce further tensile forces in the flexural reinforcement and hence cause the supports to

react horizontally. Details of the supports are shown in the left inset of Figure 2-5. Sufficient

longitudinal reinforcement was provided to prevent flexural failure from being dominant. The

nib of the dapped-ends were 180 mm long with the centroid of the support bearing plate (and

hence the reactions) located at 100 mm from the re-entrant corner. All bearing plates were 13

mm thick and a 225 mm x100 mm plan area. Figure 2-5 shows the test setup and specimen

dimensions. To measure the 45° shear displacement, displacement transducers were mounted to

the girders such that the potential failure planes emanating from the re-entrant corners were

crossed, as depicted in the right inset of Figure 2-5. The specimens were loaded using a 2 MN

capacity hydraulic actuator. To produce the loads, the actuator applied the load P onto a spreader

beam, which would then distribute the load evenly such that both loading points on the girder

would experience a force of P/2. The loading scheme consisted of a uniform load increase up to

100 kN from zero, then cyclically loading between 50 kN and 100 kN for ten cycles, and finally

loading up to failure. The purpose of the cyclic loading regime was for the bearing components

and supports to “settle in” to the specimens so as to maintain uniform contact throughout. It is

important to note that this portion did not contribute to the actual test results (Herzinger and El-

Badry 2007). The first two specimens, DE-A-0.5/1.0, were reinforced conventionally with 2-

20M horizontal bars in the nib and closely spaced stirrups near the re-entrant corner. All

specimens were loaded at half (500 mm from support) and full (1 000 mm from support) shear

spans. To provide proper anchorage, the 20M bars were welded to an L130 x 83 x 9.5 steel

angle. The next two specimens, DE-B-0.5/1.0, were reinforced with vertical headed studs and

horizontal single-headed studs, replacing, respectively, the stirrups and the horizontal 20M bars

in Specimens DE-A-0.5/1.0. Specimens DE-B-0.5/1.0 were also loaded at half and full shear

13

spans. Specimen DE-C-1.0 was reinforced with double-headed diagonal studs alone and loaded

at full shear span. The final two specimens, DE-D-1.0 and DE-D*-1.0, were reinforced with a

combination of vertical, horizontal, and diagonal studs and loaded at full shear spans (Herzinger

and El-Badry 2007).

Figure 2-5: Experimental setup and strain gauge placement for shear displacement

measurement (inset). From Herzinger and El-Badry (2007)

Figure 2-6 below illustrates detailing of the specimens. Results from the experiment are

summarized below in Table 2-1 and load-displacement graphs are shown in Figure 2-8d. The

control specimens, DE-A-0.5/1.0, had capacities, Ptest, of 421 kN and 453 kN respectively, with

critical shear cracks forming at the re-entrant corners. The test specimens had capacities ranging

from 395 kN to 528 kN. From Figure 2-7, shear cracks generally formed at the re-entrant corners

for all specimens except for DE-C-1.0, whose critical crack formed in the full-depth section

some distance away from the studs. Values of the experimental loads and those derived from

analytical procedures were nearly identical in some areas and generally within ten percent of one

14

another, while variations were as high as sixteen to eighteen percent for the “C” and “D”

specimens using shear friction. Crack widths were measured at 50 kN intervals from 100 kN to

250 kN for all specimens except DE-A-1.0, and varied between 0.88 mm and 1.94 mm at their

highest recorded values. The failure plane inclination estimated from shear friction varied

between 66° and 32°, while using diagonal bending, a much narrower range of 30°-35° was

obtained (Herzinger and El-Badry 2007).

Of particular interest were the strains in the reinforcement of Specimen DE-D*-1.0, plotted

against the applied load as illustrated in Figure 2-8a-c. In general, the headed studs were shown

to have yielded significantly well beyond its yield strain compared to the stirrups. The main

conclusion drawn in the work of Herzinger and El-Badry (2007) was that headed studs are a

viable alternative to conventional reinforcement in the design of dapped-ended girders, since the

overall amount of bars used can be reduced and additional anchorage using welded connections

with external plates or angles is not required, while at the same time, the strength would not be

compromised. Furthermore, the authors concluded that headed stud design can sufficiently be

performed using strut-and-tie models, while the shear friction and diagonal bending methods

provide an effective means to check strength capacities (Herzinger and El-Badry 2007).

Comparing the strengths predicted by the two analytical methods, however, there were

inconsistencies when comparing the “A” and “B” specimens to the “C” and “D” specimens. For

instance, there was no significant difference between the ultimate loads whether the girder failed

due to shear cracking or diagonal bending for the “A” and “B” specimens. On the other hand, the

behaviour of the “C” and “D” specimens were well predicted by diagonal bending while shear

friction overestimated their strength capacities, when compared with the experimental results. To

15

further explore the validity of the two analytical methods, the current numerical study was

therefore carried out.

16

Figure 2-6: Specimens tested by Herzinger and El-Badry (2007) (a) DE-A-0.5/1.0 (b) DE-B-

0.5/1.0 (c) DE-C-1.0 (d) DE-D-1.0 (e) DE-D*-1.0

(a)

(b)

(c)

(d)

(e)

17

Figure 2-7: Predicted vs. actual crack patterns (Herzinger and El-Badry 2007) (a) DE-A-0.5/1.0

(b) DE-B-0.5/1.0 (c) DE-C-1.0 (d) DE-D-1.0; and (e) DE-D*-1.0

(a)

(b)

(c)

(d)

(e)

18

Table 2-1: Summary of results by Herzinger and El-Badry (2007)

Figure 2-8: Specimen DE-D*-1.0 reinforcement strains (Herzinger and El-Badry 2007) (a)

Horizontal stud strains (b) Stirrup and vertical stud strains (c) Inclined stud strains

(d) Load-45° displacement curves for all specimens

(a)

(c)

(b)

(d)

19

2.3 Previous Work on Numerical Studies of Dapped-Ended Girders

2.3.1 Popescu et al. (2014)

Popescu et al. (2014), performed numerical analyses on various strengthening layouts of fibre-

reinforced polymer (FRP) systems at the re-entrant corners of dapped-ended beams using the

finite element software ATENA. Aside from the configurations tested in their original

experiments, the authors considered carbon fibre (CFRP) sheets and near-surface mounted

reinforcement (NSMR) in addition to the existing CFRP plates used. These components were

first modelled individually, placed at angles of 0°, 45°, and 90° with the horizontal. The Newton-

Raphson solution technique was employed, taking into account material and structural non-

linearity. The concrete was modelled using 8-noded plane stress elements, which had a 2x2

Gauss point layout for integration. Steel reinforcement, CFRP components, and NSMR were

modelled using 2-noded truss elements assumed to be perfectly bonded to the concrete. The

CFRP sheets and plates were hence treated as bars assigned with equivalent cross-sectional

areas. The web area of the beams was not damaged and hence a coarse mesh of 100 mm was

used, while a finer 50 mm mesh size was applied to the dapped-ends. Bond-slip laws were not

accounted for; the 2D nature of the model meant that out-of-plane de-bonding could not be

simulated, and neither could de-bonding causing tear-off of concrete. The concrete material

model considered included a fracture plastic regime which consisted of a Rankine failure

criterion with exponential softening, and fracture which was based on the crack band method.

The tensile behaviour of concrete was specified using a crack displacement approach

incorporating mode-I fracture energy. For numerical purposes, the CFRP and steel stresses were

specified at 1% of ultimate strength beyond failure to ensure stress redistribution.

20

From individual application of the components, the authors found that the 0° and 45° layouts of

the CFRP sheets, plates, and NSMR provided strength increases ranging from 6.9-23.3%. It was

also found that the 90° configurations did not result in significant improvement in capacities.

Using these results, the authors proceeded to combine all possible combinations of layouts within

the numerical study, interchanging between the strengthening regimes. Only 0° and 45° layouts

were considered since the 90° layouts provided little increase in strength. Numerical testing

showed that CFRP plate-controlled specimens (i.e. specimens with CFRP plates placed at 0°,

with the 45° component being a CFRP plate, CFRP sheet, or NSMR) increased capacity between

16.3-32.7%, while for CFRP sheet-controlled specimens the resulting strength improvement

ranged from 10.7-37.7%. Both NSMR-controlled specimens showed a 23.3% increase in

strength.

The numerical study showed that a combination of CFRP sheets used in horizontal and inclined

configurations would provide the greatest capacity increase for a dapped-ended beam, and it was

shown that CFRP systems in general are a feasible strengthening solution. From a modelling

standpoint, the limitations of the finite element models were apparent with regards to their

inability to model de-bonding behaviour, since perfect bond was assumed with no further

information available. As a result, the authors planned to perform a separate study on the bond

behaviour of FRP through investigation of strain distributions in view of plate de-bonding and

sheet de-bonding at the cracks. They also sought to investigate in greater detail the effects that

other types of FRP materials or different NSMR cross sectional dimensions would have on the

beam capacity.

21

2.3.2 Moreno and Meli (2013)

In their study, Moreno and Meli (2013) conducted testing on the strength of inverse dapped-

ended girders. The experimental objectives of the authors were to evaluate the performance of

three types of reinforcement used in dapped-ended construction, determine the load at the onset

of cracking and track their widths and propagation. The first specimen (E1) was reinforced

conventionally with the typical arrangement of vertical and horizontal hoops and stirrups, as per

PCI guidelines. The second specimen (E2) was reinforced in a manner similar to E1 except that

four 16 mm post-tensioning steel strands of ultimate strength 1862 MPa were applied with a pre-

stressing force equivalent to 74% ultimate strength to compress the re-entrant corner. The final

specimen (E3) consisted of diagonal bars placed at 45° in place of some of the conventional

reinforcement, in anticipation that they will improve crack control by tracing the tensile principal

stress path. All specimens had full-depth cross sectional dimensions of 1000 mm x 480 mm.

Overall member length was 1750 mm, while the 275 mm long nib had a depth of 250 mm.

Specimen E3 was chamfered 50 mm x 50 mm at the re-entrant corner to accommodate the

diagonal reinforcement. A 980 kN capacity load cell applied symmetrically a service load of 177

kN directly onto the dapped ends.

Following testing, the authors performed a numerical evaluation on the three specimens tested,

denoted M1, M2, and M3 corresponding to their experimental counterparts, using the discrete

and smeared crack approaches. The finite element software ANSYS was used for a three

dimensional, non-linear analysis. Cracking was simulated using zero thickness surface-to-surface

contact elements, denoted as CONTA173, by introducing a cohesive zone material which could

undergo delamination. Contact stiffnesses in each direction were updated based on average stress

between an element pair, and were modelled using a bilinear traction-separation law. Concrete

22

was modelled using the 8-noded SOLID65 element, which was capable of emulating crushing

and cracking. The Drucker-Prager failure criterion was invoked for compressive failure. Stiffness

was assumed to be lost upon crushing failure, whereas a linear softening response orthogonal to

the crack direction was adopted for tensile behaviour. Shear transfer coefficients were also

specified for crack opening and closing. Hangers, grills, bars, and pre-stressing strands were

modelled with LINK8 elements with isotropic hardening behaviour, while the remaining

reinforcement was modelled using a smeared technique. In terms of boundary conditions, the

centreline of the beam was constrained in the x and z-directions. As well, a point located 555 mm

from the end of the nib was constrained in the y-direction. Zero-length springs were applied

along the full-depth section to simulate the action of the anchorage bars which held the

specimens to the reaction slab.

The numerical results showed good correlation with experimental strength values. However,

member stiffnesses diverged slightly; in Specimen M2 the stiffness deviated from the

experimental value at an applied load of 46 kN and resulted in a displacement underestimation of

close to 15%, while for Specimen M3 the stiffnesses measured from discrete and smeared

cracking diverged from the experimental value at a 50 kN applied load and caused a 19%

overestimation. Specimen M1 showed agreement in stiffness between the smeared crack

approach and the experiment, but initial stiffness modelled by discrete cracking was different due

to an erroneous crack orientation. The initial cracking loads based on the numerical study were

accurately predicted based on smeared cracking for models M1 and M2, but only reached a value

slightly greater than two-thirds that of the experimental for Model M3. Based on the determined

crack widths using the smeared cracking approach, Model M1 showed the greatest extent of

cracking, while Model M2 displayed the smallest amount. Discrete cracking also showed that

23

M1 cracked the most with M2 cracking the least; however, the finite element results did not

correspond as accurately with the experiments beyond the numerical cracking loads.

The authors drew several key conclusions regarding their numerical study. First, it was found

that while the smeared crack model is appropriate for solving practical problems, the discrete

crack approach is best used in situations where crack directions and material parameters are

known beforehand. As well, crack widths using the discrete approach are accurate provided that

there is predominantly only one direction in which the crack propagates.

2.3.3 Argirova et al. (2014)

The study performed by Argirova et al. (2014) was a dedicated numerical examination of the

behaviour of dapped-ended beams using a finite element program developed by the authors. The

motivation behind their work was to demonstrate that non-linear analyses may adequately be

performed using simple constitutive material models which do not require a large number of

parameters whose values may affect the model sensitivity. The strategy employed is based on a

low order level-of approximation philosophy where, through the use of simple models, a

designer is not immediately overly concerned with advanced refinement (Argirova et al. 2014).

To this end, the authors used a technique known as the elastic-plastic stress field (EPSF) method.

This technique may be considered conservative and accounts for compatibility which enables the

strain state of the concrete to be represented. As preluded, a feature of this technique is its

simplicity; only the modulus of elasticity, plastic strength, and a strength reduction law

contributing to cracking are required for the complete definition. The authors have successfully

applied this method previously to deep beams, beams with openings, pre-stressed girders, frame

corners, and lightweight structures.

24

With regards to material modelling, the EPSF method assumes a Mohr-Coulomb yield surface

for concrete with no tensile capacity, and uses a strength reducing parameter to account for the

brittle nature of concrete. The concrete was treated as an elastic-perfectly plastic material whose

compressive stress strain behaviour is linear elastic up to its strength and remains constant

beyond the elastic limit. As mentioned, concrete tensile strength, and therefore tension stiffening,

were neglected. The steel was assumed to take on a bilinear form consisting of a linear elastic

portion with tensile modulus Es, and a strain hardening section with hardening modulus Eh. The

finite element mesh consisted of constant strain triangles whose principal strains could be

determined for a given displacement field. The assumption was made that principal stresses and

strains were parallel, meaning that rotation is permitted. Steel reinforcement was treated using

two-noded truss elements with dowel action ignored, and was assumed to be perfectly bonded to

the concrete. The Newton-Raphson technique was used to obtain the solution, using an open Java

program made available by the authors. The authors applied their technique to a total of thirty

dapped-ended specimens from the work of various researchers, with virtually identical

agreement between experimental and numerical failure loads. Included in their investigation

were Specimens DE-A, DE-B-1.0, and DE-D*-1.0, from Herzinger and El-Badry (2007). Using

the EPSF method, the authors found test to numerical load ratios of 0.95, 1.03, and 0.97, for

these three specimens, respectively, with cracking occurring mostly at the re-entrant corner and

significant yielding of reinforcement at these locations.

One advantage of the EPSF method is that fracture principles are not required. Furthermore, the

technique aligns well with strut-and-tie and rigid plastic stress field methods, meaning that an

exact correlation with the lower and upper bound plasticity theorems is obtained to ensure a safe

design. To summarize, Argirova et al. (2014) have shown that the EPSF method can be applied

25

accurately and conservatively to a wide range of concrete structures through simple model

definitions using a custom finite element program. With their program, the authors’ accurately

predicted the ultimate loads and failure behaviour of some of the specimens tested by Herzinger

and El-Badry (2007). However, major simplifications are inherent in their models as they are

represented in only two dimensions and contact interactions between the reinforcement and

concrete are ignored. As well, peripheral components, which include the spreader beam and

support assemblies, are neglected; given that the load-displacement behaviour of the specimens

is sought out, it is crucial for the boundary conditions to be as representative as possible to the

experiments. It is therefore worthwhile to perform a more detailed analysis in a three-

dimensional environment where the possibility of doing sensitivity analyses is not hindered by

limitations in the program.

2.4 Other Concrete Structures Modelled in ABAQUS

ABAQUS has been an attractive choice for concrete structures modelling due to its diverse range

of built-in capabilities. A number of studies have employed similar approaches for modelling

concrete, reinforcement, and shear studs, as well as their interactions. The examples presented

below are not concerned with dapped-ended girders, but they are nonetheless critical to examine

from literature the best modelling strategies to apply to the current study, such that efficiency is

not sacrificed over an unnecessarily complex treatment. Obviously, no two modelling situations

between authors will be identical, but the underlying principles may very well be applicable for a

wide range of uses.

2.4.1 Genikomsou and Polak (2014)

In their work, Genikomsou and Polak (2014) performed a non-linear analysis of slab-column

connections under static and pseudo-dynamic loading to investigate the punching shear failure

26

mode and corresponding capacities. Five specimens of either interior or edge slab-column

connections were modelled under various combinations of gravity, seismic, and unbalanced

moment loading conditions. Loads were applied in combination of linear and cyclic regimes.

Due to symmetry, only a quarter of the control and half of all other specimens were required to

be modelled. Eight-noded hexahedral elements with reduced integration (C3D8R) were used for

concrete while 2-node truss elements (T3D2) were applied to reinforcement, assumed to be

perfectly bonded to concrete by applying the embedded constraint. Restraints were introduced at

the bottom edges in the direction of the applied load, and summation of reactions at the edges

yielded the punching shear load. The control specimen was analyzed using both

ABAQUS/Standard and ABAQUS/Explicit with quasi-static conditions, while the explicit solver

was used for the remaining specimens. For all specimens a smooth step velocity was applied to

impose a desired value of central slab displacement. Concrete damaged plasticity, coupled with

fracture energy criterion and cracking displacement, was used to model concrete. Sensitivity

analyses with varying mesh sizes of 15, 20, and 24 mm were examined. For the control

specimen, dilation angles of 20, 30, 38, and 42, fracture energies of 0.07 N/mm, 0.082 N/mm,

and 0.1 N/mm, and variable viscosity parameters for the standard solver were also investigated.

Furthermore, the effect of the presence of damage parameters was also studied.

The authors concluded that the models developed can accurately predict punching shear failure

in the analyzed slab-column specimen, with the capability of being used for future parametric

studies. Their study showed that parameters such as mesh size, dilation angle, and consideration

of damage affect the accuracy of the models. The inclusion of compressive and tensile damage

resulted in an underestimation of the ultimate load since plastic strains would be lower than the

inelastic strains. However, damage had no impact on the behaviour when loading was within the

27

elastic range. Consideration of tensile damage alone in fact overestimated the failure capacity.

These observations led to the notion that compressive damage had significant impact on model

behaviour. As a result, the authors recommended that damage should not be considered in the

study of punching shear and should only be used in cyclic or dynamic loading, even if a

conservative estimate is provided. According to the authors, the issue of mesh size dependence

can be alleviated by incorporating a characteristic length in the stress-crack displacement

response as well as introducing viscoplastic regularization.

2.4.2 Demir et al. (2017)

Within this study, Demir et al. (2017) performed a numerical investigation on the effect of the

dilation angle material parameter on the behaviour of deep beams. The specimen chosen for the

study was obtained from the experiment by Roy and Brena (2008). In ABAQUS, the beam was

modelled as 3D solid 8-node linear hexahedral (C3D8R) elements, while reinforcement was

represented using 2-node linear truss (T3D2) elements. Perfect bond was assumed and therefore

the embedded technique was used to tie the reinforcement to the concrete. A mesh analysis

revealed that a 50 mm element size best optimized the model. Bearing plates were present at the

pin and roller supports, and at the point of load application. A linear vertical displacement was

applied at the loading plate. The concrete damaged plasticity model was used for concrete with

uniaxial compressive and tensile behaviour defined, along with their respective damage variables

to account for material degradation. Reinforcing steel was modelled as elastic with strain

hardening in the inelastic region. The authors performed four different analyses corresponding to

four dilation angle values selected: 20°, 30°, 40°, and 56.3°, with the final value being the

highest allowable permitted. Compared with the experimental ultimate load of 740 kN, they

28

found that a dilation angle of 56.3° produced the most accurate ultimate load of 720 kN in their

specimen, while the lowest value only yielded 450 kN.

The conclusion drawn was that the dilation angle has significant effects on the behaviour of deep

beams. The authors observed that with increasing values both strength and ductility were

increased. Because of the relatively high stiffness of deep beams, the authors recommended

using high dilation angle values of around 50° for such members, but advised that sensitivity

analyses should be carried out nonetheless such that a reasonable value can be selected.

2.4.3 Zheng et al. (2010)

The authors in this study were interested in the behaviour of concrete bridge decks supported on

longitudinal steel beams (Zheng, et al. 2010). In their finite element model, they used exclusively

shell elements (S8R or S4R) to model all components. Full connection between the concrete slab

and steel beams was assumed and imposed through the use of multi-point constraints. Variable

mesh sizes were employed to achieve a balance between strength loss due to smaller elements

and excessive prediction of energy loss due to larger elements. Steel was assumed to take on

bilinear stress-strain behaviour with classical metal plasticity and von Mises yield criterion. A

compressive power law was assigned to concrete with linear tension stiffening assumed for

simplicity. As with the previous two case studies, the concrete damaged plasticity model was

used because, based on its plastic flow and yield functions, the authors found it to be the most

suitable for their application. Both implicit/standard and explicit quasi-static solution techniques

were applied to the model.

The most important numerical observations made by the authors in this work was that an implicit

analysis produced less accurate results because stress localization occurred, whereas an explicit

29

analysis was more advantageous since it based failure on global structural behaviour.

Additionally, they noted that, while high kinetic to internal energy ratios can provide an

indication of failure, a force equilibrium check is the best method for determining whether

failure has occurred. Based on their results, a discontinuity in the matching between reactions

and applied loads corresponded to failure, whereas the kinetic to internal energy ratio may

remain low even during and beyond failure.

30

CHAPTER 3 - MODELLING APPROACH

3.1 Specimen Representation

All specimens were modelled as three dimensional components within the graphical interface of

ABAQUS (CAE). Each model consisted of the concrete girder, reinforcing cage, hinge and roller

support assemblies, and spreader beam. The spreader beam was included because the load was

applied through displacement control, which would impose different boundary conditions at both

its ends. As well, load application through displacement control means the stiffness of the

spreader beam will affect the load imparted on the girder. Most dimensions were obtained from

the original paper of Herzinger and El-Badry (2007); stud dimensions were available from a data

sheet provided by the manufacturer. Other dimensions were the top, bottom, and side clear

covers, which were 25 mm, 25 mm, and 20 mm respectively (Herzinger 2007). The two layers of

flexural reinforcement in all specimens were taken to be spaced 60 mm apart measured centre-to-

centre, as were the bottom two layers of U-shaped horizontal stirrups for specimens where they

were present. All conventional and U-shaped stirrups had an inner bend radius of 35 mm. Stirrup

hooks and overlaps were neglected. Detailed dimensions for the support assemblies and spreader

beam were obtained from the laboratory. The spreader beam was used solely to distribute loads

from the actuator to the two loading points on the concrete girders. Rollers were taken to act as

the supports for the spreader beam. In all models, the spreader beam was roughly 1.5 m long,

420 mm in depth, and had flanges slightly wider than 100 mm. The flanges and stiffeners had a

13 mm thickness while the web was 10 mm thick. Figure 3-1 shows the typical finite element

model developed to represent the specimens.

31

Figure 3-1: Overview of typical model developed in ABAQUS

The wire objects highlighted in red in Figure 3-1 did not contribute to the model, but served as

reference lines for determining the points used to measure the 45° shear displacements during

post-processing. In the output model, the node at the top end corner of the nib was selected,

followed by the node closest to where the 45° guide line crosses the girder soffit. The relative

displacements between these two nodes could hence be determined and served as the 45° shear

displacement of that end of the girder.

3.2 Meshing

ABAQUS offers an extensive library of elements for modelling three-dimensional bodies. The

major categories include solid, shell, membrane, and wire elements. Of greatest relevance in the

current study were the solid and wire elements. Examples of common solid elements of linear

order include the 8-noded hexahedral element (C3D8) and the 4-noded tetrahedron (C3D4).

Three dimensional wires can be modelled as truss elements (T3D2) or beam elements (B31)

32

(Similia 2013). A quadratic order warrants more nodes and the hexahedral element can become

10- or 20-noded depending on the element library selected, while tetrahedral elements become

ten-noded with the increased order. Many structural problems, however, can be solved accurately

without using a higher order and therefore a linear scheme was employed for all element types.

A feature of using hexahedral elements in the linear order is the ability to use reduced integration

elements (C3D8R). These elements consist of only one integration point for the calculation of

the field variable and are therefore computationally more efficient than a fully integrated element

of the same size. However, an effect known as hourglassing may arise from the use of such

elements. Since only one integration point is present, the element may deform in a manner such

as to cause zero strain at this location, which will lead to mesh distortion (Similia 2013).

ABAQUS has methods to control hourglassing, but the best option to minimize this effect is to

ensure that there are sufficient layers of elements through the thickness of the component. The

biggest advantage of using reduced integration elements is that shear locking, experienced with

fully integrated elements, can be avoided. This phenomenon arises because the formulation of

incompatible or inconsistent polynomials may lead to non-existent shear strains which will result

in overly stiff behaviour. Like hourglassing, this effect can be alleviated by using a finer mesh

(Similia 2013).

All bearing components, including the spreader beam, support assemblies, and bearing plates

were assigned C3D8R elements. Fully integrated elements, C3D8, were not used because the

effects of shear locking may yield overly stiff behaviour, even with a fine mesh size. It is

recommended that, for problems where bending and contact are present, reduced integration

elements should be used (Similia 2013).Meshing was straightforward for these parts since the

geometries were relatively simple and the nature of their meshes did not contribute significantly

33

to the final behaviour of the specimen. Figure 3-2 and Figure 3-3 show the 3D representations

and meshes of the support assemblies and spreader beam, respectively. All conventional

reinforcement and headed studs adjacent to the re-entrant corners were modelled in 3D and

meshed using C3D8R elements. Reinforcement elsewhere, including the 500 mm long 5/8”

single headed studs in the bottom of Specimens DE-B-0.5/1.0 were modelled as wire

components and meshed using B31 beam elements. Figure 3-4 shows the 3D assembly and mesh

of four specimens. Due to the radial nature of the double-headed studs, wedge elements (C3D6R)

were automatically generated near the centres of their stems.

Figure 3-2: 3D geometries (left) and meshes (right) of (a) hinge assembly; and (b) roller

assembly

(a)

(b)

34

Figure 3-3: Spreader beam 3D geometry (left) and mesh (right)

35

Figure 3-4: 3D view reinforcing cages (left) and their meshes (right) of Specimens: (a) DE-B-

0.5/1.;0 (b) DE-C-1.0; and (c) DE-D-1.0

(a)

(b)

(c)

36

Since much of the non-linear behaviour of the model arose from the concrete, mesh generation

and element section were most crucial for the girder. The difficulty of meshing depended fully

on the complexity of the geometry. To place solid components inside the girders, cavities were

required to be cut into the girders; a typical recess for accommodating studs is shown in Figure

3-5. The result was that circular, angled edges were introduced to the interior of the girder and

meshing using the conventional structured or swept techniques was not permitted. In order to

make meshing possible, the girders were partitioned until one of the above techniques not only

became applicable, but would generate an organized mesh which best minimized the use of

irregular element shapes, especially near the re-entrant corners where the cracking behaviour

may be affected by element geometry and orientation. As shown in Figure 3-6, the girders were

eventually partitioned extensively. It was found, however, that an organized mesh was not

possible even after heavy partitioning. Due to the layout of the straight and inclined studs, there

was excessive overlapping and interference of partitioning boundaries, as well as the

introduction of boundaries on adjacent partitioned cells which rendered them unsuitable for

meshing. Therefore, the portions of the girders within close proximity of the stud heads and

stirrup bends were treated as separate parts such that independent meshing could be performed.

The result was a significant improvement on the quality of the mesh of the main girder body. The

separated areas were also provided a finer mesh as can be seen in Figure 3-6, not only to improve

bearing contact between the stud heads and concrete, but also for constraint purposes, discussed

in the following section.

37

Figure 3-5: Cut view of cavity introduced into the girder of DE-B-0.5/1.0 by the ¾” single-

headed stud

An alternative for meshing the girder was to use C3D4 elements. Tetrahedral elements are robust

in the sense that mesh generation is possible even with complex geometries. However, these

types of elements are generally not recommended for stress analyses. A tetrahedral mesh

produces several times more elements when compared to a hexahedral mesh of the same side

length, which clearly has poor implications for computational efficiency. These elements can be

excessively stiff and a very fine mesh is usually required. For contact interactions, C3D4

elements are also not recommended; modified tetrahedral elements (such as C3D10M) are

preferred but, as before, computational effort is increased with their use. Based on the

aforementioned considerations, tetrahedral elements were avoided altogether.

38

Figure 3-6: Girder partition geometries (left) and typical meshes (right) of Specimens: (a) DE-B-

0.5/1.0 (b) DE-C-1.0; and (c) DE-D-1.0

3.3 Materials

3.3.1 Concrete

ABAQUS employs three main built-in models which may be used to reproduce the non-linear

behaviour of concrete. These are the smeared cracking model, brittle cracking model, and

concrete damaged plasticity (CDP) model. In the current study, the CDP model was selected not

only for its versatility, but because its features offered the most direct method of visualizing

cracking patterns. The CDP model, proposed by Lubliner et al. (1989), incorporates a yield

(a)

(b)

(c)

39

criterion, elastic-plastic strain decomposition, a flow rule, work hardening, and relates cohesion

to compressive and tensile energies released in the non-linear portions of their respective stress

strain curves through rate equations. This modelling regime may be used in both

ABAQUS/Standard and ABAQUS/Explicit solvers in models with plain or reinforced concrete

subject to monotonic, cyclic, or dynamic loadings (Similia 2013).

Yield criteria offer a basic method to describe the failure of geomaterials, including concrete, and

are described by surfaces in principal stress space. Essentially, any stress state experienced by

the material which does not lie within the surface implies failure. In the most general sense,

failure criteria take on the form:

1 2 3, , 0F I J J (3-1)

where I1, J2, and J3 are invariants of the Cauchy stress tensor (with J2 and J3 being deviatoric).

The yield criterion of the CDP model as per Lubliner et al. (1989) is given by:

2 1

13

1max maxF J I

(3-2)

with α, β, and γ being dimensionless constants. The parameters α and β are expressed in terms of

the initial equibiaxial and uniaxial compressive stresses, fb0 and fc0 respectively, and the initial

uniaxial tensile yield stress ft0 as (Lubliner, et al. 1989):

0

0

0

0

1

2 1

b

c

b

c

f

f

f

f

(3-3)

and

0

0

1 1c

t

f

f

(3-4)

40

Lubliner et al. (1989) suggest values for the stress ratio fc0/ft0 to range between 1.10 and 1.16,

based on data from experiments, which would result in α values of 0.08 to 0.12, respectively.

The constant γ is relevant only under triaxial compression, namely when the maximum principal

stress σmax is less than zero. The value of γ is dependent upon whether the stress states form

“meridians”, which can either be compressive (σ1 = σ2 > σ3) or tensile (σ1 > σ2 = σ3). The ensuing

discontinuity is accounted for by placing σmax in Macaulay brackets, where <σmax> = (|σmax| +

σmax)/2. Under triaxial compression, the compressive meridian is given by (Lubliner, et al. 1989):

2 13 3 3 1 cJ I f (3-5)

Similarly, the tensile meridian is expressed as:

2 12 3 3 3 1 cJ I f (3-6)

In both cases, fc is the maximum uniaxial compressive stress. If the ratio of the square roots of the

second stress invariants between the tensile and compressive meridians is introduced as follows:

2

2

TM

CM

J

J (3-7)

then the expression for γ becomes:

3 1

2 1

(3-8)

Values for ρ can range between 0.64 and 0.8.

The foregoing developments are of interest since they are foundational to the model used in

ABAQUS. With modifications by Lee and Fenves (1998) to incorporate cohesive strengths,

ABAQUS recasts the ensuing result as (Similia 2013):

13

1max max cF q p

(3-9)

41

where the expressions for α, β, and γ are identical to those as expressed before, except that for β,

fc0 and ft0 are replaced respectively by the effective cohesive stress components c and t (the

cohesive stress components are functions of their respective plastic strain components), and σmax

is replaced by its effective stress equivalent max . Furthermore, ABAQUS defines a second

stress invariant ratio Kc which is based on ρ from Equation (3-7). As a result, γ is simply given

by Equation (3-8) with Kc replacing ρ. The hydrostatic stress �̅� is given by (Similia 2013):

1

3p trace σ (3-10)

while the Mises equivalent effective stress �̅� is given by (Similia 2013):

3

:2

q S S (3-11)

with the effective stress deviator �̅� being (Similia 2013):

p S σ I (3-12)

The effective stress �̅� is expressed as (Similia 2013):

0 :el pl σ D ε ε (3-13)

with 𝐃0𝑒𝑙 being the initial elastic constitutive tensor, and ε and ε

pl are the total and plastic strains

respectively. Figure 3-7 and Figure 3-8 illustrate the shape of the yield surfaces viewed from two

different stress planes.

42

Figure 3-7: Yield surface viewed from biaxial principal stress plane. Adapted from ABAQUS

User’s Manual (Similia 2013)

Figure 3-8: Yield surface viewed from deviatoric stress plane, with variations in the ratio Kc.

Adapted from ABAQUS User’s Manual (Similia 2013)

43

As with other geomaterials, concrete does not remain static under sustained loading. Rather, it

flows slowly over time which is a consequence of minute changes in plastic strain as a function

of the stress state. This gives rise to a flow rule, which may generally be expressed as:

p G

εσ

(3-14)

with �̇� being a plastic multiplier and G being a potential function, whose partial derivative with

respect to the stress tensor represents the direction of the plastic strain increment orthogonal to

the surface G. An associated flow rule is one in which the potential surface G is coincident with

the failure surface F. ABAQUS, however, employs non-associated flow with the following for

the potential surface (Similia 2013):

2 2

0 tan tantG q p (3-15)

which is a Drucker-Prager hyperbolic surface, with ψ being the dilation angle at high confining

pressures in the p-q plane, and ϵ being the eccentricity. The dilation angle can vary considerably,

taking on values as low as 13° in concrete slab analysis from the ABAQUS Verification Manual

(Similia 2013), up to 56° elsewhere in confined concrete applications (Jiang and Wu 2011). The

eccentricity ϵ controls the effect that the confining pressure has on the dilation angle; a default

value of ϵ = 0.1 is specified by ABAQUS meaning that the dilation angle varies little under a

wide range of confining pressures (Similia 2013). The parameters fb0/fc0 and Kc have default

values of 1.16 and 2/3 respectively within the program, which is consistent with those originally

suggested by Lubliner et al. (1989). The plasticity component of the model includes a viscosity

parameter μ to resolve convergence issues commonplace in materials with strain softening and

stiffness degradation (Similia 2013). Its default value of zero remained unchanged for all models.

44

Constitutive models for tension and compression were obtained from existing literature to

complete the tabular data required by the program. Actual compressive and tensile strengths of

concrete were available from experimental data. In compression, the following relation was

adopted (Wahalathantri, et al. 2011):

'

'

0 @0.5

0'

,@0.5

0

, 0

,

1

c

c

c c c f

c

c

c c c endc f

c

E

f

(3-16)

where the non-linear portion is based on the model proposed by Hsu and Hsu (1994), terminating

at εc,end, corresponding to thirty percent of the concrete strength f’c. The constant β is defined as

(Wahalathantri, et al. 2011):

'

0 0

1

1 cf

E

(3-17)

with the strain ε0 at peak strength being:

5 ' 3

0 8.9 10 2.114 10 cf (3-18)

and initial tangent modulus E0 given as:

'

0 124.3 3283.12 cE f (3-19)

The above empirical expressions for strain and elastic modulus are expressed in terms of f’c with

units of ksi, and must hence be in consistent units with the compressive strengths which are

provided in MPa (Note: 1 MPa = 0.145 ksi). Figure 3-9 shows the uniaxial compression model

used with salient points labelled.

45

Figure 3-9: Compressive stress-strain curve. Adapted from Wahalathantri et al. (2011)

A strain based constitutive relationship was not used for tensile properties since mesh sensitivity

becomes an issue in regions with little reinforcement during the post peak, or tension stiffening

response portion of the tensile curve (Similia 2013). As a result, ABAQUS offers the choice of

defining tensile behaviour using a stress-displacement relationship through a fracture energy

parameter GF, proposed by Hillerborg et al. (1976). The fracture energy released is essentially

the total area under the tensile stress-displacement curve:

0

dFG f w w

(3-20)

The parameter GF must be determined experimentally. However, the following simple empirical

expression is provided in the CEB-FIP Model Code (2010) in the absence of experimental data:

0.180.073 in N/mmF cmG f (3-21)

46

where the mean compressive strength fcm (being higher than fck1 in CEB-FIP) is given by:

8 in MPacm ckf f (3-22)

A bilinear curve was employed for the strain softening portion of the stress-displacement curve

f(w) = σt (CEB-FIP 2010):

1

1

1

1

1 0.8 ,

0.050.25 ,

ctm

t

ctm c

wf w w

w

wf w w w

w

(3-23)

with the concrete tensile strength fctm provided from experimental values, where the crack widths

(displacements) w1 and wc are given respectively by:

1F

ctm

Gw

f (3-24)

and

5 Fc

ctm

Gw

f (3-25)

From Figure 3-10, the adopted tensile-displacement curve takes on the following shape and the

fracture energy is, physically speaking, the shaded area under the curve.

1 fck is the characteristic strength used for analysis and design in the CEB-FIP code, hence it was taken as f’c

47

Figure 3-10: Tension-displacement curve for concrete. Adopted from CEB-FIP (2010)

Damage evolution is one of the key features of the CDP model. In a nutshell, elastic and plastic

degradation parameters are introduced into the constitutive matrix in order to decrease the

material stiffness. This may be expressed as (Lubliner, et al. 1989):

1 1, ; , pd σ D ε ε (3-26)

where d1,… are the elastic degradation variables and δ1,… are the plastic degradation variables.

ABAQUS specifies the following (Similia 2013):

01 :el pld σ D ε ε (3-27)

where the damage variable d can take on values between 0 and 1, with zero indicating no damage

and one being fully damaged. In terms of user input, ABAQUS requires solely the uniaxial

constitutive behaviour of concrete in tension and compression; therefore, Equation (3-27) may be

reduced to:

01 pl

i i i id E (3-28)

48

where the subscript i can denote tension or compression. The stress-strain (or stress-

displacement) curves, as well as damage parameters, must be specified as a function of inelastic

plastic strains in both cases. The effective cohesive stresses, as brought up earlier, is therefore:

0

pl

i i iE (3-29)

which will specify the size of the yield surface. ABAQUS requires inelastic strains to be entered

as a tabular function of stress. The program then converts the inelastic strains into plastic strains

which is dependent on the damage variables as follows:

01

pl in i ii i

i

d

d E

(3-30)

where 𝜀�̃�𝑖𝑛 is the inelastic crushing strain, 𝜀�̃�

𝑖𝑛, in compression, and cracking strain, 𝜀�̃�𝑐𝑘, in

tension. Each value of plastic strain is checked whether it is decreasing and/or negative, in which

case an error message will appear. The above was applied to compression only since, as

introduced, a displacement-based constitutive model was employed for the tensile regime. Figure

3-11 shows how ABAQUS incorporates compressive damage into the constitutive model and

gives the definition of inelastic and plastic strains.

49

Figure 3-11: Definition of inelastic and plastic strains in compression (Similia 2013)

In tension, if a stress-displacement relationship is specified, ABAQUS converts the user supplied

displacement values into plastic equivalents, namely:

0

01

pl ck t tt t

t

d lu u

d E

(3-31)

with l0 being the specimen length defaulted by the program as unity. The inelastic strain is the

difference between the total strain and the strain taken as if the material were still within the

elastic range:

0

in ii i

E

(3-32)

For simplicity, an expression was adopted for compressive and tensile damage whereby

degradation was measured as a direct proportion of the strength remaining in the concrete in the

strain softening branch; in other words:

1 ii

i

df

(3-33)

50

The above relationship was adopted for compressive damage in some studies such as in Yu et al.

(2008). However, an expression of this type would not be valid prior to reaching the compressive

strength since, in compression, concrete leaves its elastic range well before f’c is reached. As a

result, the above is applicable only in the post peak region and attains a maximum value

corresponding to the proportion of strength remaining at the termination of the compressive

curve. Some damage models seek out an expression for damage that is a function of inelastic

strain. However, existing models, such as the one developed by Alfarah et al. (2017) consist of

parameters which require the use of a characteristic length associated with mesh element size.

This is undesirable since damage is assumed to be a material property independent of mesh

characteristics. Other examples of damage expressions, such as that used by Tao et al. (2014),

are of little use since they are essentially a rearrangement of Equation (3-30), which signifies an

iterative dependence between damage and plastic strain. The method of Wahalathantri et al.

(2011) specifies damage evolution as a ratio between inelastic strain and total strain, but

substitution of the resulting values into Equation (3-30) lead to zero plastic strains everywhere

along the curve, which is prohibited by ABAQUS. In the absence of damage, ABAQUS will

treat the model as purely plastic, since Equation (3-30) becomes:

pl in

i i (3-34)

and Equation (3-31) reduces to:

pl ck

t tu u (3-35)

The concept of crack formation through an integration point cannot be realized using the CDP

option. However, crack directions can be visualized by enabling maximum principal strains in

the field output, whose vectors would indicate that cracks would have formed in the direction

perpendicular to them (Similia 2013). Another method of observing cracking behaviour would

51

be to enable the tensile damage field output variable. In this option, cracks may be associated

with the extent which the material is damaged. Table 3-1 below lists the compressive and tensile

concrete strengths for each specimen mix to be used in all material definitions.

Table 3-1: Concrete properties (Herzinger 2007)

Specimen f’c (MP(a) fct (MP(a)

DE-A-0.5 38.0 3.45

DE-A-1.0 38.1 4.27

DE-B-0.5 36.9 3.67

DE-B-1.0 38.6 4.04

DE-C-1.0 41.9 3.95

DE-D-1.0 36.8 3.77

DE-D*-1.0 39.2 3.75

The density of concrete in all specimens was taken as 2.4x10-9

tonnes per cubic millimetre2. An

initial Poisson’s ratio of ν = 0.2 was assigned, but, as will be seen and explained in Chapter 5, a

value of zero was eventually adopted, which is valid for cracked concrete.

3.3.2 Steel

From Figure 3-12, an idealized bilinear stress-strain model with hardening was adopted for all

steel components, with ultimate conditions dictating the terminus of the curve. Actual values of

elastic modulus, yield stress, and ultimate stress and strain for all conventional reinforcement and

studs were obtained from the stress-strain data provided in the experiments. All conventional bars

were found to have minimum yield strengths of 430 MPa. No information on the bearing steel

was provided, therefore standard values of Es = 200 000 MPa, fy = 400 MPa, and fu = 550 MPa

2 In ABAQUS, the consistent units used for length, force, mass, time, stress, energy, and density were mm, N, tonne,

s, MPa, mJ, and tonne/mm3, respectively.

52

were assigned. As well, a hardening modulus Eh of 2 000 MPa, or one percent of Es, was assumed

for the bearing steel. From elementary theory, the yield strain is given by:

y

y

s

f

E (3-36)

and Eh determines the ultimate strain with:

u y

u y

h

f f

E

(3-37)

The hardening portion of the curve is entered in ABAQUS as a plastic property. Experimental, or

engineering values of stresses and strains are expressed as nominal values. The program requires

the data tabulated as true stress and strain. The two stress and strain types are related through:

1tru nom nom (3-38)

with:

ln 1tru nom (3-39)

Hence, the required plastic strains are expressed as:

trupl tru

sE

(3-40)

Figure 3-12: Actual (left) vs. adopted (right) stress strain curve for mild and headed stud steels

53

Similar to the conventional reinforcement, material data for all studs were obtained from the

experimentally generated stress-strain plots. The single-headed studs had yield strengths ranging

from 597 and 633 MPa, while those of the double headed studs ranged between 492 and 544

MPa. The 5/8” double headed studs, however, had yield strength of 364 MPa. The elastic

modulus and ultimate stress for the studs were again specified from testing (Herzinger and El-

Badry 2007). Table 3-2 below summarizes the key properties of all steel components.

Table 3-2: Steel properties (Herzinger 2007)

Reinforcement

Type Size fy (MP(a) fu (MP(a) εu

* Es (MP(a)

Bearing Steel

All angles,

support

assemblies, and

bearing plates

400 550 0.077**

200 000

Conventional

reinforcing bars

10M 430 687 0.12 189 898

20M 501 664 0.14 205 246

25M 461 657 0.12 191 531

Single-headed

studs

3/8” 597 679 0.035 198 969

1/2” 626 693 0.03 201 622

5/8” 618 682 0.035 195 749

3/4” 633 700 0.032 192 894

Double-headed

studs

1/2” 492 582 0.087 192 087

5/8” 364 549 0.17 183 765

3/4” 544 607 0.045 198 177 *Interpreted from experimental stress-strain diagrams

**Based on Eh = 2 000 MPa

Since data for the conventional reinforcement and studs were available, it was not necessary to

define a hardening modulus for these components. A Poisson’s ratio of ν = 0.3 and density of

7.85x10-9

tonne/mm3 were applied universally for all steel. The yield strength of steel is typically

54

taken at a point measured at a 0.2% offset from the strain corresponding to the point where the

stress-strain curve is no longer linear. For simplicity, this was neglected in the material models.

3.4 Contact Interactions and Constraints

The choice of representing the majority of the reinforcement as 3D wire elements arose from the

need to maintain efficiency while minimizing complexity of the model in defining

reinforcement-to-concrete interaction. Conventional reinforcement and headed studs away from

the re-entrant corners were treated as perfectly bonded to the concrete through the use of the

embedded technique, which was assumed to be valid since these bars were deformed. In this

case, the concrete was treated as the host element while the bars were the embedded elements.

Undoubtedly, a solid representation of the entire reinforcing cage would vastly increase the

difficulty of meshing the girder, since more cavities would be introduced and required to be

accommodated during partitioning. Moreover, the increased number of elements, and either

specifying tie constraints to simulate perfect bond between the interfaces, or defining bond-slip

behaviour, would increase computational cost. The tension stiffening property defined in the

CDP model provides a simple and approximate means of accounting for the bond interaction

between steel and concrete (Similia 2013). Double headed studs, however, could not be assumed

to have perfect bond since their stems were plain. Instead, there is cohesive action between the

stud-concrete interface, which was modelled using traction-separation behaviour. All

conventional reinforcement and studs passing through the re-entrant corner were modelled as 3D

solids such that contact interactions between the steel and concrete may be defined, which is not

possible using wire elements. Figure 3-13 illustrates the contact surfaces for reinforcement to

concrete interaction.

55

Figure 3-13: Interacting surfaces, highlighted in red, specified between the inner concrete

surfaces created by the studs (top) and the studs themselves (bottom). Specimen

DE-D-1.0 shown

In ABAQUS, cohesive interaction may be modelled using cohesive elements or specified as a

property. For simplicity, the latter method was chosen. Three stiffness coefficients are required

for input – namely, a normal stiffness Knn, and two components of tangential stiffness Kss and Ktt

– as well, a bond-slip relationship was required which expressed the stiffness in terms of a

maximum stress and corresponding displacement, or slip. The model code CEB-FIP 2010

specifies that, for plain bars, the maximum shear stress τmax for which slip is imminent may be

related to the strength of concrete through:

56

'0.3max cf (3-41)

while for deformed bars:

'2.5max cf (3-42)

Hence, the tangential stiffness components can, in simplest terms, be expressed as (Henriques, et

al. 2012):

1

maxss ttK K

s

(3-43)

where s1 is the relative interface slip taken as 0.1 mm and 1.0 mm for smooth and deformed bars,

respectively (CEB-FIP 2010). The deformed stems of the single-headed studs consisted of

indentations as opposed to ribs; the former would exhibit lower bond strength. However, due to

lack of experimental data, and because the above expressions for shear stress and slip do not

differentiate between the type of deformation used, the single-headed studs were assumed to

have the same level of interaction with the concrete as the conventional reinforcement. For plain

bars, τmax = τf and s1 = s2 = s3, corresponding to complete failure of the bond-slip interaction

without plateauing or softening once the peak stress and corresponding displacement are

reached. Looking at Figure 3-14 (left), typical bond-slip laws between steel and concrete consist

of a non-linear ascending branch, peak plateau, linear descending branch, and lower plateau. In

ABAQUS, however, traction-separation behaviour can only be approximated using a bilinear

model such as that shown in Figure 3-14 (right). Since not more than several millimetres of slip

were expected between the steel and concrete, and in order to capture the peak plateau, an

arbitrarily large failure displacement of 1000 mm was assigned for all single-headed studs and

stirrups passing through the re-entrant corners whose bars were deformed. No cohesive action in

the normal direction was assumed in any of the cohesive properties specified. In addition to

57

cohesive interaction, a standard normal-tangential contact was assigned between all 3D

reinforcement and concrete. The default “hard” pressure-overclosure property was specified for

normal contact, while a static exponential decay formulation was defined for tangential friction.

An arbitrarily large decay coefficient of 1E10 was specified in the absence of experimental data

to ensure that any slipping motion present would transition quickly from static to kinetic states of

friction, whose coefficients were taken as 0.6 and 0.45 respectively.

Figure 3-14: Typical bond-slip law for reinforcement in pullout (left) and traction separation

model available in ABAQUS (right)

The steel angles in Specimens DE-A-0.5/1.0 were assumed fully bonded to the concrete and

hence tied to the girder. The welded connections between the 20M bars and the angles were

accounted for by merging the two parts together as one. The interaction between the components

of the roller support consisted of the following:

Roller head to roller bearing plate

Roller bearing plate to roller cart wheels

Roller cart wheels to roller base

Additionally, contact was assigned between the sides of the roller cart wheels and the side of the

roller frame itself. For the above, all interactions were considered steel-to-steel with “hard”

58

normal contact and tangential exponential decay friction, with a large decay value of 1E10 as

before. Static and kinetic coefficients of friction were respectively taken as 0.74 and 0.57.

A kinematic coupling constraint was assigned to the roller wheels such that, where permitted,

rolling would occur either on the roller base or roller bearing plate. Steel-to-steel interaction was

also specified between the hinge roller and hinge head, while the hinge roller was tied to the

hinge base. Figure 3-15 shows the interacting contact surfaces of the support assemblies. Lastly,

steel-to-steel interaction was applied to contact between the spreader beam rollers and bearing

plates on the top face of the concrete girder, with the contacting surfaces shown in Figure 3-16.

Figure 3-15: Interacting surfaces highlighted in red, for the roller (left) and hinge (right)

support assemblies

59

Figure 3-16: Interacting surfaces, highlighted in red, between spreader beam support roller and

girder load bearing plate. The roller cylinder is truncated at the connection to the

spreader beam to enable a definite, flat surface to be present in order for the roller

to be tied to the bearing plate above. ABAQUS cannot tie surfaces which are

tangent to one another.

Tie constraints were generated between the roller head and hinge head with their respective

dapped concrete faces. Thin areas overlapped by the support faces were neglected to avoid

potential mesh distortion, as shown in Figure 3-17. The bearing plates on the top of the girder

were also tied to the concrete. When applying tie constraints, it is recommended for the slave

surface to consist of a softer material and finer mesh compared to the master surface. As a result,

all the steel components were taken to be the master surface while the concrete was treated as the

slave surface for the tie constraints.

60

Figure 3-17: Thin strips of area belonging to the girder overlapped by the support faces were

ignored when applying the tie constraints, shown by the yellow arrows

As stated previously, the sections of concrete girder within close proximity of the stud heads

were modelled as separate parts. To combine them with the main girder body, tie constraints

were applied between these “concrete heads”, shown in red in Figure 3-18, and the main girder

body. The surfaces on the girder to which these heads were attached were treated as the master

surface, while the surfaces of the heads themselves were given slave assignments. The tied

surfaces are shown highlighted in Figure 3-18. Hence, the concrete heads required a finer mesh

than the main girder body. This was important considering that both interacting entities have the

same material. Also, a constraint tolerance of 0.1 mm was specified. It was found that if the

concrete heads had a similar or larger mesh size to the main girder body, and no tolerance was

assigned, the mesh at these locations could distort uncontrollably. All contact interactions were

assigned using the general contact option in ABAQUS. No pure master-slave surface pairs were

assigned and therefore all interacting surfaces were treated with equal master-slave weighting, in

61

other words, no penetration of master nodes to slave nodes would occur. Surface-to-surface

formulation was selected for all tie constraints.

Figure 3-18: The sections of the girders in red were treated as separate parts and tied to the

main girder body, shown by the highlights: (a) DE-B-0.5/1.0; (b) DE-C-1.0; and

(c) DE-D-1.0

3.5 Loads and Boundary Conditions

All specimens were modelled using z-symmetry about a plane bisecting the long direction of the

model. From Figure 3-19, this corresponded to the plane whose normal was the z-axis. This

boundary condition was applied to the concrete girder, spreader beam, supports, and stirrups

whose planes or nodes met with the plane of symmetry. Specimens DE-C-1.0 and DE-D/D*-1.0,

however, were not symmetrical. In the former, there is overlap between the diagonal studs

(a) (b)

(c)

62

causing one to be placed slightly above the other, while the latter have a 3/8” horizontal stud

placed on only one side of the vertical stud. To enforce symmetry for Specimen DE-C-1.0, the

diagonal stud was moved far enough over so as to be entirely on one side of the plane of

symmetry, and the stud was extended slightly to compensate for the small amount of length lost

to overlapping. For Specimens DE-D/D*-1.0, as mentioned previously, only half of the middle

horizontal 3/8” stud was modelled. From Figure 3-20, a boundary condition of zero movement in

the z-direction was applied to the cut face of the studs to simulate the effect of the presence of

the other half of the stud being reflected about the plane of symmetry. Fixed boundary conditions

were enforced on the base of the hinge and roller supports to prevent their movement.

Load control was applied to the specimens in the experimental program. Nonetheless, the force

exerted on the girder is due to the actuator pushing down on the specimen, and the specimen

resisting the action through its stiffness. Specifying an applied load on the spreader beam

corresponding to some point near or beyond the predicted ultimate conditions resulted in failure

of the analysis to complete; severe mesh distortion was the cause because the model did not have

the ability to sustain the maximum load specified. This asymptotic behaviour meant that post-

peak response and plateaus in the load-displacement curves could not be captured using load

control. As a result, a displacement boundary condition was applied on the actuator bearing plate

on top of the spreader beam, which in turn transferred reaction forces, acting as point loads, onto

the girder. The displacement assigned was 25 mm downwards in all the models, which was

applied in a smooth step function on the bearing plate at the top of the spreader beam. The plate

was also constrained from moving in the longitudinal direction.

63

Figure 3-19: z-symmetry boundary condition applied to the central plane of girders

Figure 3-20: Zero z-displacement boundary condition applied to 3/8” half stud

64

3.6 Solution Technique and Analysis Steps

For a general class of static stress analysis problems in the finite element method, the objective is

to solve governing equilibrium equations to obtain the deformed state of the specimen based on

the constitutive properties of the material and its reaction to applied loads. In their most basic

form, these equations are expressed as

Ku p (3-44)

where K is the stiffness matrix, u is a vector of nodal displacements, and p is the vector of

applied forces. K and p are the summations of all N individual element (e) stiffnesses and loads,

namely (Rao 2011)

1

Ne

e

K K (3-45)

1

Ne

e

p p (3-46)

with

( )

T

e

e

V

dV K B DB (3-47)

In Equation (3-47), B describes the state of strain based on the interpolating shape function

matrix N. The shape functions are selected by the user and may be in the form of polynomials or

sinusoidal expressions. B and N are related as follows (Rao 2011):

65

0 0

0 0

0 0

0

0

0

x

y

z

y x

z y

z x

B N (3-48)

and D is the constitutive matrix. The vector p is the amalgamation of all concentrated loads pc,

loads due to initially deformed states pi, surface tractions ps, and body forces pb. This can be

written for each element as:

e e e e e

c i s b p p p p p (3-49)

where

( )

T

0 e

e

i

V

dV p B Dε (3-50)

( )

T e

e

s

S

dS p N Φ (3-51)

( )

T e

e

b

V

dV p N (3-52)

and ε0, Φ, and ϕ are the initial strain, surface traction, and body force vectors, respectively. With

K and p known, Equation (3-44) can be solved to obtain the nodal displacements as follows

1u K p (3-53)

but not before boundary conditions are applied, otherwise K remains singular and cannot be

inverted. Hence, from the above, stress and strain states can be described through constitutive

66

definitions. The above provides the fundamental basis for the standard solution technique within

any typical finite element program. In most cases, it is computationally heavy to invert K and

numerical methods are used to solve for u. This being said, for non-linear problems, Newton’s

method is commonly employed within ABAQUS. By considering the equilibrium condition,

Equation (3-44), to be an expression of the general form3

0n mF u (3-54)

small adjustments 𝛥𝑢𝑖+1𝑚 are applied to the current solution at increment i such that Equation

(3-54) becomes (Similia 2013)

1 0n m m

i iF u u (3-55)

Expanding the above into a Taylor approximation the following series is acquired:

2

1 1 1 0

n m n m

i in m p p q

i i i ip p q

F u F uF u u u u

u u u

(3-56)

Ignoring higher order terms by assuming a small value of 1

m

ic , this expression reduces to:

1

np p n

i i iJ u F (3-57)

with:

n

np ii p

FJ

u

(3-58)

being the Jacobian matrix (which is related to the tangent stiffness matrix), and:

n n m

i iF F u (3-59)

Hence, the next approximation is given by:

1 1

m m m

i i iu u u (3-60)

3 Equations (3-54) - (3-60) are expressed in index notation. The superscripts m, n, p, and q are not exponents, but

denote the entry of the vector or matrix in question. This notation is adopted by ABAQUS Theory Manual and

enables partial derivatives to be expressed with little difficulty.

67

where 1

m

iu can now be expressed in terms of the inverted Jacobian from Equation (3-58), and

iterations are performed until n

iF and 1

m

iu contain entries deemed small enough, at which point

convergence is achieved. The Jacobian matrix may be difficult or impossible to acquire in closed

form, and it must be solved in every iteration. Furthermore, inversion of the Jacobian might be

laborious or impossible. As a result, Newton’s method is not attractive for large problems. Other

modified variations of the method are possible in which the Jacobian is not obtained at every

increment. Of particular interest are highly non-linear structures, which add complexity to the

stiffness, constitutive, and hence Jacobian matrices. Newton’s method is typically best suited for

linear problems with small deformations and simple elastic materials.

A standard, general static procedure was initially used to perform the simulations within the

current study. However, it was found that even small applied displacements of fractions of a

millimetre were too sensitive for the solver to proceed; the size of increment required was

excessively small and the job would abort prematurely. The presence of concrete meant that

material non-linearity was inevitable, with the difficulty of the problem being compounded by

the presence of contact interactions and number of elements present. Clearly, it was not possible

or feasible to continue using the general static technique. To overcome this difficulty, an explicit

solver was used instead to perform the analyses. The equation of motion in a dynamic analysis

takes on the general form

0 Mu Cu Ku (3-61)

where M, C, and K are the mass, damping, and stiffness matrices, respectively. The formulation

of the solution technique is based on an explicit central time difference integration, expressed at

time increment i as (Similia 2013):

68

1 1 1

2 2

2

i ii i

it t

u u u (3-62)

and

1

1 1 2i

i i it

u u u (3-63)

where u, �̇�, and �̈�, are displacement, velocity, and acceleration vectors respectively. The

acceleration at any increment is given by (Similia 2013)

1i i i u M p I (3-64)

where p(i)

is the vector of applied loads and I(i)

is the vector of inertial forces. A major advantage

of the explicit scheme is that the mass matrix M is diagonal. Furthermore, inversion of the

stiffness matrix and iterations are not necessary for the solver to proceed. As a result, problems

with non-linear materials and large deformations can be modelled. The explicit solver is

generally applied to dynamic models where high velocity events such as impact are present.

However, the applicability of this method to static stress analyses is equally as valid; such a

problem is known as a quasi-static analysis. In essence, the loads are applied slowly enough such

that the inertial forces I are zero. Hence, in a quasi-static analysis, Equation (3-64) reduces to

1i iu M p (3-65)

To confirm the validity of a quasi-static analysis, equilibrium must be checked manually by

ensuring that all components with imposed loads or boundary conditions have balancing reaction

forces. Unlike the standard scheme, the explicit solver does not perform equilibrium checks

automatically. Moreover, assuming all components are rigid bodies, a general rule of thumb is

that the kinetic energy of the model must not exceed 5% of the total internal energy during

analysis. In many cases this requirement is not satisfied at the very beginning of analyses since

small amounts of movement will be present before deformations due to applied loads occur

69

(ABAQUS 2005). To maintain quasi-static conditions, the loads were applied slowly enough

such that the energy restriction was satisfied. Applying the loads too quickly will introduce

significant inertial effects; in the case of the specimens this meant that the load bearing plate

would sink into the material, causing local bulging and no bending deformations would result as

it should. Stability of the analysis is measured based on the stable time increment, which is

defined as:

e

d

Lt

c (3-66)

where Le is the smallest characteristic length for an element, and cd is the material dilational

wave speed of the corresponding element given by:

d

Ec

(3-67)

with E and ρ being the elastic modulus and density, respectively. A smaller time increment is

indicative of longer analysis time required. Due to slow loading in a quasi-static analysis,

millions of time increments may be required and computational time can reach the order of

hundreds of hours. Running the model in its natural time period is often impractical, which is

problematic for situations where rate dependency is present (ABAQUS 2005). As a result,

ABAQUS enables users to artificially increase the stable time increment, and hence reduce

analysis time. In rate dependent problems, this is best achieved through mass scaling. From

Equation (3-67), it is apparent that if the material density is increased by some amount f2, the

stable time increment would increase by the amount f. This specific method is known as fixed

mass scaling and is typically applied to the entire model at the start of the analysis step.

Excessive amounts of mass scaling can increase inertial effects and care must be taken to ensure

70

quasi-static conditions are retained. Other mass scaling methods are available but will not be

covered in detail here.

Two steps were defined for the analysis: initial and load. All support and symmetry boundary

conditions were applied in the initial step. The displacement was applied during the load step. A

recommendation was made that the time period should be at least 10 times the largest frequency

of the model for loading. A suitable time period of 3 seconds was selected through trial and error

to ensure quasi-static conditions. As mentioned previously, the displacement was applied over a

smooth step amplitude. ABAQUS defines the smooth step using the following third order

polynomial

3 2

1

1 1 1

10 15 6i i ii i i

i i i i i i

t t t t t ta A A A

t t t t t t

(3-68)

such that the amplitude a, which varies between 0 and 1, is Ai at ti, and Ai+1 at ti+1. A mass

scaling factor of 16 was applied at the beginning of the step for the entire model to decrease

analysis time by roughly four times.

71

CHAPTER 4 - RESULTS OF SENSITIVITY ANALYSES

4.1 Sensitivity Analyses

All seven specimens were initially run at least three to four times corresponding to the different

sensitivity analyses performed. Specimens DE-A-0.5, DE-B-0.5, DE-C-1.0, and DE-D-1.0 were

selected as the reference cases, where the variation of mesh size, concrete dilation angle, and the

presence of damage were the primary factors studied which would be anticipated to affect the

behaviour of the models. Specimens DE-A-1.0, DE-B-1.0, and DE-D*-1.0 will herein be referred

to as the remaining specimens. Specimens DE-A-1.0 and DE-B-1.0 were not analyzed for the

effect of mesh size or the inclusion of damage because, except for their shear spans, all of their

geometries and reinforcing layouts were identical to those of their respective half shear span

counterparts. Likewise, Specimen DE-D*-1.0 was not analyzed for mesh size nor damage

because of its similarity in geometry and reinforcement configuration to Specimen DE-D-1.0.

The mesh size and damage regime used by the remaining specimens was adopted from the best

values shown by their respective reference specimens. The best dilation angle values from the

reference specimens were applied to the corresponding remaining specimens. Specimen DE-D-

1.0 was the only model where a further examination of dilation angle was not initially performed

because of good agreement between the numerical and the experimental behaviour based on the

original value chosen. In all reference cases, specified mesh sizes of 7, 10, 15, and 20 mm were

adopted for the main concrete girder body. All other components’ mesh sizes were held constant

throughout the mesh size variations. In ABAQUS, individually seeding edges with a finer mesh

will also create a fine mesh through the depth of the model in order for the meshed regions to be

compatible. This meant that attempting to refine the mesh at the re-entrant corners would also

cause the other parts of the girder to be refined. The complex geometry as a result of partitioning

72

and presence of cavities in the girder body exacerbated this problem. Biased meshing towards

the re-entrant corners was therefore avoided. The geometry indicated that more than one million

elements would be generated if biased meshing were to be applied properly. Furthermore, as will

be discussed later, a finer mesh may not be indicative of a more accurate model given the current

model definitions. A dilation angle value of 35°, with both compressive and tensile concrete

damage included, was first assumed and applied throughout the mesh analysis. The best mesh

size was chosen to examine the effect of including damage in three cases: (1) both compressive

and tensile damage are considered (2) only tensile damage is considered; or (3) no damage is

included. Finally, the best damage regime, coupled with the mesh size previously chosen, was

used to perform the dilation angle sensitivity, where values ranging from 35° to 55° were

examined. As will be shown in the next chapter, additional studies were performed after the

sensitivity analyses to further improve the numerical results. All other parameters in the CDP

model were fixed throughout, and no other material model for either steel or concrete was

investigated. Contact interactions were also kept constant for all models in every study. Due to

the nature of the models it was not possible to measure the crack widths at any point in the

analyses. In all cases it was desired to keep numerical ultimate loads to within 10% of

experimental values. To obtain the applied load from the output models, the vertical reaction

experienced at the spreader plate, where the vertical displacement was directly applied, was

determined and multiplied by two, since only half of the specimens were modelled. The 45°

shear displacements were calculated by measuring the difference between the initial and final

distances of two points on the girder soffit along a 45° line starting from the top corner of the nib

and ending at the full depth. Because of the 500 mm depth of the specimens this would amount

to 707.1 mm initial distance between the two points. Due to imperfections in the element node

73

alignments, this value deviated slightly from specimen to specimen, but in all models the small

difference was neglected.

Failure of the specimens mostly took place before completion of the analyses. In the following

sections, the load-displacement curves are shown in dotted lines beyond numerical failure up to

the experimental shear displacement in the cases where numerical failure occurred early.

4.2 Results of the Sensitivity Analyses

This section presents the results from the main sensitivity analyses for both the reference and

remaining models. Grouping of the specimens with regards to the presentation of their results in

subsequent sections of this chapter will be based on the type of sensitivity analyses performed, as

opposed to geometric similarity. Since results from the reference models affected the choice of

mesh size, dilation angles, and damage regime used for the remaining models, the former will be

presented first.

4.2.1 Reference Models

4.2.1.1 Specimen DE-A-0.5

The mesh analysis for DE-A-0.5 revealed some minor differences in ultimate load and the 45°

displacement at failure between the four assigned element sizes in the pre-failure state. From

Figure 4-1, in all mesh cases, the maximum loads varied from 337 to 364 kN, corresponding to

underestimations between 25.6 to 24.5 percent when compared with 453 kN from the

experiment. The high shear ductility of the dapped end shown in the experiments was almost

entirely absent; while the 45° displacement of the specimen actually reached 20 mm, the mesh

cases showed a ductility of slightly greater than 10 percent of that value at most. From the

analysis, a mesh size of 15 mm was selected, and a damage sensitivity test was next performed.

74

For variation of the damage regime, as seen in Figure 4-2, there was no significant difference in

load-displacement behaviour between the two models where full damage and tension damage

only was considered; in the consideration of tension damage only the capacity was 330 kN and

failed at 2.6 mm shear displacement. However, in the case where the model was treated as purely

plastic (i.e. no damage), the ultimate load increased to 391 kN and shear displacement reached

12.0 mm. In this case it is important to note that maximum load was reached after the assumed

point of failure, but at that point in the analysis massive plastic straining had engulfed the re-

entrant corner so failure was taken to be at an earlier point. For the dilation angle analysis, shear

displacements increased with the increase in the dilation angle. At 45° dilation angle, the

specimen failed after displacing 7.4 mm, while for 50° dilation angle, the displacement reached

prior to failure was 14.4 mm. At 55° dilation angle, no clear point of failure was identified even

though significant damage had propagated from the re-entrant corners at both ends. Typically,

failure did not occur until cracking spread to the rest of nib. Table 4-1 summarizes the sensitivity

analysis results for Specimen DE-A-0.5.

75

Table 4-1: Summary of results of sensitivity analyses of Specimen DE-A-0.5

Parameter

Girder

Mesh Size

(mm)

Dilation

Angle

Damage

Regime

Numerical

Load

Capacity

(kN)

Numerical

Shear

Displacement

at Failure

(mm)

Failure

Location

(End)

Girder

Mesh Size

7

35° Full

346 2.1

Re-entrant

corner

(Hinge)

10 364 2.5

15 337 2.5

20 343 2.5

Damage

Regime 15 35°

Tension

Only 330 2.6

Re-entrant

corner

(Hinge)

No Damage 391 12.0

Re-entrant

corner

(Roller)

Dilation

Angle 15

45°

Tension

Only

457 7.4

Re-entrant

corner

(Hinge)

50° 468 14.4

Re-entrant

corner

(Roller)

55° 484

Did not

completely

fail

Re-entrant

corner

(Hinge)

76

Figure 4-1: Specimen DE-A-0.5 mesh analysis (full damage, ψ=35°)

Figure 4-2: Specimen DE-A-0.5 damage regime analysis (mesh size=15 mm, ψ=35°)

0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25 30

Ap

plied

Lo

ad

(kN

)

45° Displacement at Failing End (mm)

Mesh Size=7 mm

Mesh Size=10mm

Mesh Size=15mm

Mesh Size=20mm

Experimental

0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25 30

Ap

plied

Lo

ad

(kN

)


Full Damage

Tensile DamageOnly

No Damage

Experimental

77

Figure 4-3: Specimen DE-A-0.5 dilation angle analysis (mesh size=15 mm, tension damage only)

4.2.1.2 Specimen DE-B-0.5

Noticeable differences were present in load-displacement behavior of DE-B-0.5 in variation of

the mesh sizes under pre-failure conditions, especially between the results shown by 7 and 10

mm compared with 15 and 20 mm. This is observed in Figure 4-4. Maximum loads were again

grossly underestimated, as in DE-A-0.5, and in the most extreme case was underestimated by

close to 40%. Displacements also showed some improvement with finer mesh size, but as before

the models could not capture the 7.7 mm shear displacement actually exhibited. Values ranged

from 2.5 mm predicted by the 20 mm model to 4.0 mm predicted by the 10 mm model. A mesh

size of 15 mm was chosen to carry over to the damage regime study, which, as can be seen in

Figure 4-5, showed little difference between the full and tension damage only models.

Considering tension damage only, the dilation angle analysis was performed, and from Figure

4-6, higher dilation angle values resulted in larger load capacities and greater ductility.

Increasing the dilation angle up to the near-maximum value of 55° resulted in a 364 kN load

0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25 30

Ap

plied

Lo

ad

(kN

)


DilationAngle=35°

DilationAngle=45°

DilationAngle=50°

DilationAngle=55°

Experimental

78

capacity and displacement of 5.8 mm at failure. Table 4-2 summarizes the results for Specimen

DE-B-0.5.

Table 4-2: Summary of results of sensitivity analyses of Specimen DE-B-0.5

Parameter

Girder

Mesh Size

(mm)

Dilation

Angle

Damage

Regime

Numerical

Load

Capacity

(kN)

Numerical

Shear

Displacement

at Failure (mm)

Failure

Location

(End)

Girder

Mesh Size

7

35° Full

331 3.6

Re-entrant

corner

(Roller)

10 326 4.0

15 276 2.9

20 255 2.5

Damage

Regime 15 35°

Tension Only 273 2.6 Re-entrant

corner

(Roller) No Damage 325 8.8

Dilation

Angle 15

45°

Tension Only

316 3.9 Re-entrant

corner

(Roller)

50° 322 4.5

55° 366 5.8

Figure 4-4: Specimen DE-B-0.5 mesh analysis (full damage, ψ=35°)

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12

Ap

plied

Lo

ad

(kN

)


Mesh Size=7 mm

Mesh Size=10 mm

Mesh Size=15 mm

Mesh Size=20 mm

Experimental

79

Figure 4-5: Specimen DE-B-0.5 damage regime analysis (mesh size=15 mm, ψ=35°)

Figure 4-6: Specimen DE-B-0.5 dilation angle analysis (mesh size=15 mm, tension damage only)

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12

Ap

plied

Lo

ad

(kN

)


Full Damage

Tensile DamageOnly

No Damage

Experimental

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12

Ap

plied

Lo

ad

(kN

)


Dilation Angle=35°

Dilation Angle=45°

Dilation Angle=50°

Dilation Angle=55°

Experimental

80

4.2.1.3 Specimen DE-C-1.0

The numerical models showed little difference in strength and ductility of Specimen DE-C-1.0 in

the mesh analysis as illustrated by Figure 4-7. However, for a mesh size of 7 mm the shear

displacement exceeded 9 mm prior to failure, while the 10 mm mesh size model showed that

failure at the re-entrant corner occurred before any significant damage propagated in the full

depth portion of the girder contrary to the failure mode observed in the experiment. Ultimate

loads, which varied from 575 kN for a 15 mm mesh size and 568 kN for a 20 mm mesh size,

were well predicted and failure was otherwise shown to occur in the full depth portion of the

girder as seen in the experiment. Despite significant failure taking place at the full depth in most

cases, the load-displacement graphs did not capture failure points as distinctively as a specimen

whose failure occurred at the re-entrant corner. Consequently, the shear displacements were

overestimated and the specimen continued to function despite exceeding the experimental shear

displacement value of 3.6 mm. Failure points were therefore taken as the points in which full

depth damage ceased to propagate. As in the two previous specimens, a mesh size of 15 mm was

selected for the damage analysis. The first two regimes involving damage were very similar in

load-displacement behaviour; however, the tension damage only case showed re-entrant corner

failure and a 591 kN ultimate load, exceeding the experimental value by nearly 12%. Likewise,

the absence of damage resulted in over-estimations of strength, which exceeded 600 kN, and the

ductility was also excessive at close to 9 mm with failure occurring at the re-entrant corner.

Despite showing deviating results, the tension only damage scheme was chosen, out of

consistency with the previous specimens, to perform the dilation angle analysis, in expectation

that the behaviour would change based on the value selected. As observed in Figure 4-8, this

assumption proved to be true; a reduction in dilation angle to 30° lowered the strength capacity

81

to 553 kN and directed failure back to the full depth portion of the girder. No other dilation angle

values were investigated as the single reduction was deemed satisfactory. Table 4-3 summarizes

results for Specimen DE-C-1.0.

Table 4-3: Summary of results of sensitivity analyses of Specimen DE-C-1.0

Parameter

Girder

Mesh Size

(mm)

Dilation

Angle

Damage

Regime

Numerical

Load

Capacity

(kN)

Numerical Shear

Displacement at

Failure (mm)

Failure

Location

(End)

Girder

Mesh Size

7

35° Full

551 9.8 Full depth

(Hinge)

10 558 4.4

Re-entrant

corner

(Roller)

15 575 4.8 Full depth

(Hinge)

20 568 4.2 Full depth

(Hinge)

Damage

Regime 15 35°

Tension

Only 591 4.8 Re-entrant

corner

(Roller) No Damage 603 9.7

Dilation

Angle 15 30°

Tension

Only 553 4.1

Full depth

(Hinge)

82

Figure 4-7: Specimen DE-C-1.0 mesh analysis (full damage, ψ=35°)

Figure 4-8: Specimen DE-C-1.0 damage regime analysis (mesh size=15 mm, ψ=35°)

0

50

100

150

200

250

300

350

400

450

500

550

600

0 2 4 6 8 10 12

Ap

plied

Lo

ad

(kN

)


Mesh Size=7 mm

Mesh Size=10 mm

Mesh Size=15 mm

Mesh Size=20 mm

Experimental

0

50

100

150

200

250

300

350

400

450

500

550

600

650

0 1 2 3 4 5 6 7 8 9 10

Ap

plied

Lo

ad

(kN

)


Full Damage

Tensile Damage Only

No Damage

Experimental

83

Figure 4-9: Specimen DE-C-1.0 dilation angle analysis (mesh size=15 mm, tension damage only)

4.2.1.4 Specimen DE-D-1.0

In terms of ultimate loads predicted, the mesh size nalysis for Specimen DE-D-1.0 produced very

consistent values, but whether the specimen failed during the analyses was dependent on the

mesh size, as highlighted in Figure 4-10. Failure and shear displacements showed some

differences depending on the mesh size considered; for mesh sizes of 7 and 20 mm, for example,

the specimen did not fail. Shear displacements were underestimated with respect to the

experimental value of 21.7 mm in the specimens which did fail; ranging from 11.2 mm for the 10

mm mesh to 14.4 mm for the 15 mm mesh. The 15 mm mesh size was chosen for the other two

sensitivity studies similar to before. As illustrated in Figure 4-11, consideration of tensile

damage only and no damage resulted in lack of specimen failure. It was determined that the

applied vertical displacement on the spreader plate needed to be increased from its original value

of 25 mm. A 40 mm displacement was hence introduced as a trial value, imposed upon the

model with a 15 mm mesh size, 40° dilation angle, and tension damage only. The result,

0

50

100

150

200

250

300

350

400

450

500

550

600

0 1 2 3 4 5 6

Ap

plied

Lo

ad

(kN

)


Dilation Angle=30°

Dilation Angle=35°

Experimental

84

highlighted in Figure 4-12, was that the ultimate load remained nearly the same as before, as

expected, but with the increased displacement failure clearly taking place at the hinged end of the

specimen. The shear displacement was also accurately predicted at 21.0 mm prior to specimen

failure. Table 4-4 summarizes the results of the base studies for Specimen DE-D-1.0. No dilation

angle studies were performed for this specimen because it was determined that the default value

provided acceptable results.

Figure 4-10: Specimen DE-D-1.0 mesh analysis (full damage, ψ=35°)

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Ap

plied

Lo

ad

(kN

)


Mesh Size=7 mm

Mesh Size=10 mm

Mesh Size=15 mm

Mesh Size=20 mm

Experimental

85

Figure 4-11: Specimen DE-D-1.0 damage regime analysis (mesh size=15 mm, ψ=35°)

Figure 4-12: The applied displacement for Specimen DE-D-1.0 was increased from 25 mm to 40

mm in order for failure to occur (mesh size=15 mm, ψ=35°, tension damage only)

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Ap

plied

Lo

ad

(kN

)


Full Damage

Tensile Damage Only

No Damage

Experimental

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Ap

plied

Lo

ad

(kN

)


Applied Displacement=25 mm

Applied Displacement=40 mm

Experimental

86

Table 4-4: Summary of results of sensitivity analyses of Specimen DE-D-1.0

Parameter

Girder

Mesh

Size

(mm)

Dilation

Angle

Damage

Regime

Applied

Displace-

ment

(mm)

Numerical

Load

Capacity

(kN)

Numerical

Shear

Displacement

at Failure

(mm)

Failure

Location

(End)

Girder

Mesh Size

7

35° Full 25

431 Did not fail

completely

Re-entrant

corner

(Roller)

10 423 11.2

Re-entrant

corner

(Roller)

15 409 14.8

Re-entrant

corner

(Hinge)

20 403 Did not fail

completely

Re-entrant

corner

(Roller)

Damage

Regime 15 35°

Tension

Only 25

418 Did not fail

completely

Re-entrant

corner

(Hinge) No

Damage 431

Applied

Displace-

ment

15 35° Tension

Only 40 424 21.0

Re-entrant

corner

(Hinge)

4.2.2 Remaining Models

From the results presented in Section 4.2.1, all the reference models predicted fairly accurate

ultimate loads using a concrete girder body average mesh size of 15 mm with tensile damage

only considered. From the dilation angle studies, the values of 50°, 55°, 30°, and 35° for

Specimens DE-A-0.5, DE-B-0.5, DE-C-1.0, and DE-D-1.0, respectively, produced the best

results. Based on these results, the aforementioned selected parameters from the 0.5 m shear span

specimens were applied to their full shear span counterparts for the dapped-ended Specimens A

and B. Likewise, the parameters from DE-D-1.0 were applied to DE-D*-1.0.

87


The analysis of DE-A-1.0 was performed using identical parameters to DE-A-0.5. However, the

model was highly unstable and excessive mesh distortion was unavoidable in localized elements

at one of the re-entrant corners. Attempts were made to modify the mesh size for the concrete

region around the stirrup bends, which as mentioned before were meshed separately. In almost

all cases where unwanted mesh distortion occurred a simple mesh size change for the concrete

regions around the solid reinforcement ends sufficed, but such was not the case for DE-A-1.0.

The dilation angle was increased slightly to values between 50° and 55°, but proved to be

unsuccessful. As observed in Figure 4-13, the load clearly reached a plateau region but well

before failure was even approached. Table 4-5 summarizes the additional dilation angle study

results performed on Specimen DE-A-1.0.

Table 4-5: Summary of results of sensitivity analyses of Specimen DE-A-1.0

Girder

Mesh Size

(mm)

Dilation

Angle

Damage

Regime

Numerical

Load

Capacity

(kN)

Numerical

Shear

Displacement

at Failure

(mm)

Failure

Location

(End)

15 50° Tension

Only 425

Did not

completely

fail

Re-

entrant

corner

(Roller)

88

Figure 4-13: Specimen DE-A-1.0 results (mesh size=15 mm, ψ=50°, tension damage only)


Based on the same parameters used for DE-B-0.5, the numerical model for DE-B-1.0 predicted a

peak load of 361 kN. The shear displacement was poorly estimated at 5.8 mm when compared to

11.8 mm from the experiment. The default model analysis failed at 52%, but right after failure of

the specimen was reached. The mesh size around the vertical stud head regions was increased

slightly to 11 mm, which improved the load capacity to 403 kN and resulted in a shear

displacement of 8.5 mm at failure. The analysis in this case failed at 69% completion. Both cases

are shown in Figure 4-14. Table 4-6 summarizes the results from the dilation angle studies.

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12

Ap

plied

Lo

ad

(kN

)


Numerical

Experimental

Analysis failed at 44%

89

Table 4-6: Summary of results of sensitivity analyses of Specimen DE-B-1.0

Girder

Mesh Size

(mm)

Dilation

Angle

Damage

Regime

Adjustment:

Concrete

Vertical Stud

Head Region

Mesh Size

(mm)

Numerical

Load

Capacity

(kN)

Numerical

Shear

Displacement

at Failure

(mm)

Failure

Location

(End)

15 55° Tension

Only

10 378 5.8

Re-entrant

corner

(Roller)

11 403 8.5

Re-entrant

corner

(Hinge)

Figure 4-14: Specimen DE-B-1.0 sensitivity results (mesh size=15 mm, tension damage only, ψ=55°)

4.2.2.3 Specimen DE-D*-1.0

The behaviour of DE-D*-1.0 was very similar to DE-D-1.0. From the latter, it was predicted that

the default 25 mm applied displacement would not be sufficient in causing failure of the

specimen. Therefore, in all cases a 40 mm displacement was imposed. The behaviour is depicted

in Figure 4-15. Applying parameters identical to those applied to DE-D-1.0 resulted in a

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12 14

Ap

plied

Lo

ad

(kN

)


Concrete Vertical Stud Head Region - Mesh Size=10 mm


Experimental



90

maximum load of 428 kN and a shear displacement of 13.6 mm, a low but conservative

underestimation when compared with the experimental value of 19.4 mm. Failure occurred at the

re-entrant corner from the roller end. Table 4-7 summarizes the results for DE-D*-1.0.

Table 4-7: Summary of results of sensitivity analyses of Specimen DE-D*-1.0

Girder

Mesh

Size

(mm)

Dilation

Angle

Damage

Regime

Applied

Displacement

(mm)

Numerical

Load

Capacity

(kN)

Numerical

Shear

Displacement

at Failure

(mm)

Failure

Location

(End)

15 35° Tension Only 40 429 13.6

Re-entrant

corner

(Roller)

Figure 4-15: Specimen DE-D*-1.0 results (mesh size=15 mm, tension damage only, ψ=35°)

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Ap

plied

Lo

ad

(kN

)


Numerical

Experimental

91

CHAPTER 5 - RESULTS FROM ADDITIONAL STUDIES

5.1 General

The overall results for the specimens analyzed in Chapter 4 showed general agreement in

behaviour between the finite element models and the experiments. However, the analyses for

Specimens DE-A-1.0 and DE-B-1.0 failed prematurely due to uncontrolled element distortion,

despite numerous attempts to refine the models. Especially considering that a 15 mm mesh size

was not considered overly fine, the elements should be at least more stable than if a smaller mesh

size was used. It was found that modification of the concrete Poisson’s ratio from the initially

assigned value of 0.2 to zero resolved the issue. All seven specimens were therefore re-analyzed

with a concrete Poisson’s ratio of zero, with all the selected parameters unchanged. The

Poisson’s ratio of cracked concrete is zero because the presence of discontinuities will cause one

region of the material to behave separately from another; homogeneity is lost and lateral

deformation cannot be sustained. Overall behaviour of the specimens remained similar to before.

In this chapter, the reinforcement strains and failure patterns are presented for the best case

selected.

5.1.1 Reference Specimens


Changing the concrete Poisson’s ratio from 0.2 to zero did not affect the overall specimen

behaviour by a large amount. The ultimate load was only increased by 3 kN while the shear

displacement at failure increased by slightly less than a millimetre to 15.3 mm. The updated

behaviour is illustrated in Figure 5-1 and the results are summarized in Table 5-1. From Figure

5-2 to Figure 5-4, the stirrups adjacent to the re-entrant corner yielded significantly, and the 20M

92

reinforcement also exceeded the yield point by a considerable amount. Stirrup strains dropped

rapidly further away from the re-entrant corner, contributing almost nothing to the strength of the

specimen. The average strains in the flexural reinforcement between the two ends were very

similar and did not exceed 250 microstrain. Due to the relatively short length of the beam, there

was very little bending present, observed by examining the vertical displacement of the beam

soffit at the different load levels in Figure 5-6.

Table 5-1: Summary of Specimen DE-A-0.5 results from Poisson’s ratio study

Girder

Mesh

Size

(mm)

Dilation

Angle

Damage

Regime

Poisson’s

Ratio

Numerical

Load

Capacity

(kN)

Numerical

Failing Shear

Displacement

(mm)

Failure

Location

(End)

15 50° Tension

Only

0.2 468 14.4 Re-entrant

corner

(roller) 0 471 15.3

Figure 5-1: Effect of concrete Poisson’s ratio on Specimen DE-A-0.5 (mesh size=15 mm, tension

damage only, ψ=50°)

0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25

Ap

plied

Lo

ad

(kN

)


Poisson's Ratio=0.2

Poisson's Ratio=0

Experimental

93

Figure 5-2: Specimen DE-A-0.5 stirrup strains at east end

Figure 5-3: Specimen DE-A-0.5 stirrup strains at west end

0

50

100

150

200

250

300

350

400

450

500

0 5000 10000 15000 20000 25000 30000 35000 40000 45000

Ap

plied

Lo

ad

(kN

)

Microstrain

Stirrup 1, East (Num.)



Stirrup 1, East (Exp.)



0

50

100

150

200

250

300

350

400

450

500

0 5000 10000 15000 20000 25000 30000 35000 40000

Ap

plied

Lo

ad

(kN

)

Microstrain

Stirrup 1, West (Num.)



Stirrup 1, West, (Exp.)

Stirrup 2, West (Exp.)


94

Figure 5-4: Specimen DE-A-0.5 20M horizontal reinforcement strains

Figure 5-5: Specimen DE-A-0.5 average flexural reinforcement strains

0

50

100

150

200

250

300

350

400

450

500

0 2000 4000 6000 8000 10000 12000

Ap

plied

Lo

ad

(kN

)

Microstrain

Horizontal Bar, East (Num.)

Horizontal Bar, West (Num.)

Horizontal Bar, East (Exp.)

Horizontal Bar, West (Exp.)

0

50

100

150

200

250

300

350

400

450

500

0 100 200 300 400 500 600

Ap

plied

Lo

ad

(kN

)

Microstrain

Flexural, East (Num.)

Flexural, West (Num.)

Flexural, East (Exp.)

Flexural, West (Exp.)

95


different load levels. Dotted lines show experimental values

Failure patterns are highlighted in Figure 5-7. At numerical failure, it is seen that tensile damage

was smeared over a large region from the re-entrant corner and even extended into the full-depth

area of the girder near the re-entrant corner. However, the initial orientation and form of the

cracking is better visualized at peak load as illustrated in Figure 5-7c where it is apparent that, as

in the experiment, cracking first occurred and propagated from the re-entrant corner. At

numerical failure corresponding to the experimental shear displacement at failure, the damage is

further increased at the nib, especially in the region immediately above the steel angle. At this

point, however, the reinforcement ceased to engage in the finite element model and the specimen

was considered ineffective.

0

2

4

6

8

10

12

14

16

Δy, W1 Δy, W2 Δy, CL Δy, E2 Δy, E1

y-D

isp

lacem

en

t (m

m)

Location Along Bottom Fibre

100 kN

200 kN

300 kN

400 kN

450 kN

453 kN

96

Fig

ure

5-7

: Spec

imen

DE

-A-0

.5 f

ail

ure

patt

erns

at

(a)

pea

k lo

ad;

(b)

nu

mer

ical

fail

ure

; (c

) nu

mer

ical

fail

ure

corr

espondin

g t

o

exper

imen

tal

shea

r dis

pla

cem

ent

at

fail

ure

; and (

d)

exper

imen

tal

fail

ure

(a)

(b)

(c)

(d)

97


The load-displacement behaviour of Specimen DE-B-0.5 was not affected dramatically by

changing the concrete Poisson’s ratio; decreasing the value to zero increased the ultimate load by

22 kN to 388 kN while the shear displacement remained nearly identical at 5.4 mm, as seen in

Figure 5-8. From the studies of Specimen DE-B-1.0 involving the aborted analyses, it was found

that increasing the mesh size slightly in the concrete surrounding the vertical stud heads

improved the strength and ductility of the specimen. Under this assumption, the same adjustment

was carried for Specimen DE-B-0.5, and as expected, the maximum strength improved to

403 kN while the shear displacement surpassed the experimental value, reaching 10.3 mm prior

to failure. This result is shown in Figure 5-9.

Table 5-2 summarizes the updated findings. Both vertical studs yielded with the east stud

yielding to a greater extent than the west one, corresponding well to the fact that failure occurred

at the hinged end. Conversely, the horizontal studs failed to engage just before reaching yielding.

The stirrups immediately beyond the vertical studs contributed nothing to the strength of the

specimen based on the strains they experienced; as well, the flexural reinforcement failed to

exceed 100 microstrain. Reinforcement strains for Specimen DE-B-0.5 are illustrated in Figure

5-10 to Figure 5-13. Similar to Specimen DE-A-0.5, the vertical deflections between the ends

and quarter points of the girder were indicative that very little bending was present at the load

levels, seen in Figure 5-14.

98

Table 5-2: Summary of Specimen DE-B-0.5 results from additional studies

Girder

Mesh

Size

(mm)

Dilation

Angle

Damage

Regime

Poisson’s

Ratio

Adjustment:

Concrete

Vertical Stud

Head Region

Mesh Size

(mm)

Numerical

Load

Capacity

(kN)

Numerical

Shear

Displacement

at Failure

(mm)

Failure

Location

(End)

15 55° Tension

Only

0.2 10

366 5.8 Re-

entrant

Corner

(Hinge) 0

388 5.4

11 403 10.3

Figure 5-8: Effect of concrete Poisson’s ratio on Specimen DE-B-0.5 (mesh size=15 mm, tension

damage only, ψ=55°)

0

50

100

150

200

250

300

350

400

450

0 1 2 3 4 5 6 7 8 9

Ap

plied

Lo

ad

(kN

)


Poisson's Ratio=0

Poisson's Ratio=0.2

Experimental

99

Figure 5-9: Effect of mesh size of concrete surrounding vertical stud heads on Specimen DE-B-

0.5 (mesh size=15 mm, tension damage only, ψ=55°, ν=0)

Figure 5-10: Specimen DE-B-0.5 vertical stud strains

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12

Ap

plied

Lo

ad

(kN

)




Experimental

0

50

100

150

200

250

300

350

400

450

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Ap

plied

Lo

ad

(kN

)

Microstrain

Vertical Studs, East (Num.)

Vertical Studs, West (Exp.)

Vertical Studs, East (Exp.)


100

Figure 5-11: Specimen DE-B-0.5 horizontal stud strains

Figure 5-12: Specimen DE-B-0.5 stirrup strains

0

50

100

150

200

250

300

350

400

450

0 1000 2000 3000 4000 5000 6000 7000

Ap

plied

Lo

ad

(kN

)

Microstrain

Horizontal Studs, East (Num.)

Horizontal Studs, West (Num.)

Horizontal Studs, East (Exp.)

Horizontal Studs, West (Exp.)

0

50

100

150

200

250

300

350

400

450

0 200 400 600 800 1000 1200 1400 1600 1800

Ap

plied

Lo

ad

(kN

)

Microstrain





101

Figure 5-13: Specimen DE-B-0.5 average flexural reinforcement strains


different load levels. Dotted lines show experimental values

0

50

100

150

200

250

300

350

400

450

0 50 100 150 200 250

Ap

plied

Lo

ad

(kN

)

Microstrain





0

1

2

3

4

5

6

7

8

9

10


Dis

pla

cem

en

t (m

m)


100 kN

200 kN

300 kN

400 kN

402 kN

102

From Figure 5-15, failure of Specimen DE-B-0.5 was shown by the numerical models to occur at

the re-entrant corner. Because the ultimate capacity of the specimen was reached at almost the

same time as the experimental shear displacement, the failure patterns the two cases were nearly

identical as seen in Figure 5-15a and c. The numerical failure occurred at a greater shear

displacement than in the experiment, therefore tensile damage at that point, shown by Figure

5-15d, was slightly heavier.

103

Fig

ure

5-1

5:

Spec

imen

DE

-B-0

.5 f

ail

ure

patt

erns

at

(a)

pea

k lo

ad;

(b)

num

eric

al

fail

ure

; (c

) nu

mer

ical

fail

ure

corr

espondin

g t

o

exper

imen

tal

shea

r dis

pla

cem

ent

at

fail

ure

; and (

d)

exper

imen

tal

fail

ure

(a)

(b)

(c)

(d)

104

5.1.1.3 Specimen DE-C-1.0

Specimen DE-C-1.0 remained largely similar in behaviour with the change in Poisson’s ratio.

However, in the case with Poisson’s ratio equal to 0.2, failure eventually took place at the re-

entrant corner at the roller end of the girder. Nonetheless, considerable damage occurred in the

full depth well before dapped-ended failure occurred and the point of failure of the specimen was

taken as the point in which no further increase in damage at the full depth was observed. The

updated results are shown in Figure 5-16 and summarized in Table 5-3.

Table 5-3: Summary of Specimen DE-C-1.0 results from additional studies

Girder

Mesh

Size

(mm)

Dilation

Angle

Damage

Regime

Poisson’s

Ratio

Numerical

Load

Capacity

(kN)

Numerical

Shear

Displacement at

Failure (mm)

Failure

Location

(End)

15 30° Tension

Only

0.2 553 4.1 Full

depth

(Hinge) 0 547 4.4

Figure 5-17 to Figure 5-21 show reinforcement strains for Specimen DE-C-1.0. Because failure

occurred in the full depth portion of the girder, the inclined stud strains at the re-entrant corners

remained below yielding, reaching not more than 2 500 microstrain before ceasing to engage.

The stirrups also remained below yielding and did not engage beyond 2 500 microstrain.

However, the horizontal hair pin in the nib at the west end reached 12 000 microstrain despite no

failure occurring at that end. While flexural strains typically remained very small, the bottom

25M bar at the failing end strained to more than 3 000 microstrain which corresponded well to

the fact that failure occurred in close proximity to that region in the full depth girder portion.

Figure 5-22 illustrates the vertical deflection of the girder at various load levels. The failure

patterns are shown in Figure 5-23. As in the experiment, principal cracking initiated in the full

depth of the girder while the re-entrant corner remained largely intact. Note that, as shown by

105

Figure 5-23d, in the experiment the crack at failure occurred in the soffit, whereas in the finite

element model damage cracking propagated from the side of the girder. Cracking at numerical

failure was undoubtedly more severe than at the shear displacement corresponding to

experimental failure, as can be seen in Figure 5-23b and c.

Figure 5-16: Effect of concrete Poisson’s ratio on Specimen DE-C-1.0 (mesh size=15 mm,

tension damage only, ψ=30°)

0

50

100

150

200

250

300

350

400

450

500

550

600

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

Ap

plied

Lo

ad

(kN

)


Poisson's Ratio=0.2

Poisson's Ratio=0

Experimental

106

Figure 5-17: Specimen DE-C-1.0 stirrup strains at east end

Figure 5-18: Specimen DE-C-1.0 stirrup strains at west end

0

50

100

150

200

250

300

350

400

450

500

550

600

650

0 1000 2000 3000 4000 5000 6000 7000

Ap

plied

Lo

ad

(kN

)

Microstrain







0

50

100

150

200

250

300

350

400

450

500

550

600

650

0 500 1000 1500 2000 2500 3000 3500

Ap

plied

Lo

ad

(kN

)

Microstrain







107

Figure 5-19: Specimen DE-C-1.0 U-stirrup strains

Figure 5-20: Specimen DE-C-1.0 inclined stud strains

0

50

100

150

200

250

300

350

400

450

500

550

600

650

0 2000 4000 6000 8000 10000 12000 14000

Ap

plied

Lo

ad

(kN

)

Microstrain

U-Stirrup, East (Num.)

U-Stirrup, West (Num.)

U-Stirrup, East (Exp.)

U-Stirrup, West (Exp.)

0

50

100

150

200

250

300

350

400

450

500

550

600

650

0 500 1000 1500 2000 2500 3000

Ap

plied

Lo

ad

(kN

)

Microstrain

Inclined Studs, East (Num.)

Inclined Studs, West (Num.)

Inclined Studs, East (Exp.)

Inclined Studs, West (Exp.)

108

Figure 5-21: Specimen DE-C-1.0 average flexural reinforcement strains

Figure 5-22: Specimen DE-C-1.0 vertical deflection at ends and quarter points of full depth at

different load level. Dotted lines show experimental values

0

50

100

150

200

250

300

350

400

450

500

550

600

650

0 500 1000 1500 2000 2500 3000 3500

Ap

plied

Lo

ad

(kN

)

Microstrain


Bottom Bar, East (Num.)



Bottom Bar, East (Exp.)


-2

0

2

4

6

8

10

12

14

16


Dis

pla

cem

en

t (m

m)


100 kN

200 kN

300 kN

400 kN

450 kN

500 kN

528 kN

109

Fig

ure

5-2

3:

Spec

imen

DE

-C-1

.0 f

ail

ure

patt

ern

s at

(a)

pea

k lo

ad;

(b)

num

eric

al

fail

ure

; (c

) nu

mer

ical

fail

ure

corr

espo

ndin

g t

o

exper

imen

tal

shea

r dis

pla

cem

ent

at

fail

ure

; and (

d)

exper

imen

tal

fail

ure

(a)

(b)

(c)

(d)

110

5.1.1.4 Specimen DE-D-1.0

Changing Poisson’s ratio of concrete proved to have little effect on the behaviour of Specimen

DE-D-1.0; the ultimate load remained at 418 kN while the displacement at failure dropped

slightly to 17.5 mm. Figure 5-24 depicts these results. The descending branch of the

experimental load-displacement curve was still not captured, however, despite reasonable

strength and ductility estimations. The updated findings are summarized in Table 5-4. Figure

5-24 to Figure 5-30 illustrate the reinforcement strains of Specimen DE-D-1.0. All the headed

studs exceeded yielding which shows that they were used efficiently. Both vertical and inclined

studs exceeded 20 000 microstrain. Failure of the specimen corresponded to the point at which

the last stud was no longer engaged, shown by the east inclined stud which surpassed 35 000

microstrain at failure. The horizontal studs did not yield to as great an extent as the vertical or

inclined studs, reaching at most around 6 000 microstrain before they stopped engaging. Strains

in the stirrups contributed little to strength and did not reach yielding. The flexural reinforcement

strains were the smallest and peaked at slightly over 250 microstrain. Figure 5-31 shows the

vertical displacement at different load levels for Specimen DE-D-1.0.

Table 5-4: Summary of Specimen DE-D-1.0 results from additional studies

Girder

Mesh

Size

(mm)

Dilation

Angle

Damage

Regime

Applied

Displacement

(mm)

Concrete

Poisson’s

Ratio

Numerical

Load

Capacity

(kN)

Numerical

Shear

Displacement

at Failure (mm)

Failure

Location

(End)

15 35° Tension

Only 40

0.2 424 21.0 Re-entrant

corner

(Hinge) 0 418 17.5

111

Figure 5-24: Effect of concrete Poisson’s ratio on Specimen DE-D-1.0 (mesh size=15 mm,


Figure 5-25: Specimen DE-D-1.0 vertical stud strains

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Ap

plied

Lo

ad

(kN

)


Poisson's Ratio=0

Poisson's Ratio=0.2

Experimental

0

50

100

150

200

250

300

350

400

450

0 5000 10000 15000 20000 25000 30000 35000

Ap

plied

Lo

ad

(kN

)

Microstrain

Vertical Stud, East (Num.)

Vertical Stud, West (Num.)

Vertical Stud, East (Exp.)

Vertical Stud, West (Exp.)

112

Figure 5-26: Specimen DE-D-1.0 inclined stud strains

Figure 5-27: Specimen DE-D-1.0 horizontal stud strains

0

50

100

150

200

250

300

350

400

450

0 5000 10000 15000 20000 25000 30000 35000 40000

Ap

plied

Lo

ad

(kN

)

Microstrain

Inclined Stud, East (Num.)

Inclined Stud, West (Num.)

Inclined Stud, East, (Exp.)

Inclined Stud, West (Exp.)

0

50

100

150

200

250

300

350

400

450

0 1000 2000 3000 4000 5000 6000 7000

Ap

plied

Lo

ad

(kN

)

Microstrain





113

Figure 5-28: Specimen DE-D-1.0 stirrup strains at east end

Figure 5-29: Specimen DE-D-1.0 stirrup strains at west end

0

50

100

150

200

250

300

350

400

450

0 200 400 600 800 1000 1200 1400 1600 1800

Ap

plied

Lo

ad

(kN

)

Microstrain





0

50

100

150

200

250

300

350

400

450

0 200 400 600 800 1000 1200 1400

Ap

plied

Lo

ad

(kN

)

Microstrain





114

Figure 5-30: Specimen DE-D-1.0 average flexural reinforcement strains

Figure 5-31: Specimen DE-D-1.0 vertical deflection at ends and quarter points of full depth at


0

50

100

150

200

250

300

350

400

450

0 100 200 300 400 500 600 700 800 900 1000

Ap

plied

Lo

ad

(kN

)

Microstrain





0

2

4

6

8

10

12

14

16


y-D

isp

lacem

en

t (m

m)


100 kN

200 kN

300 kN

400 kN

415 kN

115

The cracking behaviour of Specimen DE-D-1.0 is shown in Figure 5-32. Multiple cracks formed

and propagated from the re-entrant corner, shown at peak load in Figure 5-32a. Figure 5-32b

shows that at numerical failure, the tensile damage corresponding to the crack branch closest to

the re-entrant corner increased, until it nearly engulfed the nib at the shear displacement at

experimental failing, seen in Figure 5-32c. While the crack behaviour was predicted well by the

models, the severe damage which occurred at the top of the girder in the experiment was not

captured.

116

Fig

ure

5-3

2:

Sp

ecim

en D

E-D

-1.0

fail

ure

patt

ern

s at

(a)

pea

k lo

ad;

(b)

num

eric

al

fail

ure

; (c

) nu

mer

ical

fail

ure

corr

espo

ndin

g t

o

exper

imen

tal

shea

r dis

pla

cem

ent

at

fail

ure

; and (

d)

exper

imen

tal

fail

ure

(a)

(b)

(c)

(d)

117

5.1.2 Remaining Specimens

The original decision to use a concrete Poisson’s ratio of zero instead of the initial value of 0.2

stemmed from the fact that analyses for the full-depth A and B specimens failed extensively due

to excessive mesh distortion. Applying the change resulted in successful completion of the

analyses, but as in the reference specimens the behaviour of the remaining specimens remained

largely similar.


The dilation angle of 50° selected for Specimen DE-A-1.0 was based on the good agreement it

provided for Specimen DE-A-0.5. As predicted, the results using the chosen dilation angle value

zero Poisson’s ratio were in good agreement with experimental behaviour, with the maximum

load being 427 kN and shear displacement at failure being 10.2 mm. These results are

summarized in Figure 5-33 and

Table 5-5. In general, the stirrups adjacent to the re-entrant corner yielded significantly; more so

at the failing end whose maximum strain nearly reached 25 000 microstrain. The 20M

reinforcement at the failing end yielded, but to a lesser extent than the stirrups; failure of the

specimen occurred after this component no longer engaged. Conversely, the 20M reinforcement at

the intact end only approached yielding. The third stirrups from either end contributed very little

to the specimen strength; likewise, the flexural strains were very small and in the range of 100-

160 microstrain. The vertical girder deflection at various load levels is plotted in Figure 5-38.

118

Table 5-5: Summary of Specimen DE-A-1.0 results from additional studies

Girder

Mesh

Size

(mm)

Dilation

Angle

Damage

Regime

Concrete

Poisson’s

Ratio

Numerical

Load

Capacity

(kN)

Numerical Shear

Displacement at

Failure (mm)

Failure

Location

(End)

15 50° Tension

Only

0.2 425 Did not

completely fail

Re-entrant

corner

(Hinge)

0 427 10.2

Re-entrant

corner

(Roller)

Figure 5-33: Effect of concrete Poisson’s ratio on Specimen DE-A-1.0 (mesh size=15 mm,


0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12

Ap

plied

Lo

ad

(kN

)


Poisson's Ratio=0

Poisson's Ratio=0.2

Experimental

119

Figure 5-34: Specimen DE-A-1.0 stirrup strains at east end

Figure 5-35: Specimen DE-A-1.0 stirrup strains at west end

0

50

100

150

200

250

300

350

400

450

0 2000 4000 6000 8000 10000 12000 14000 16000

Ap

plied

Lo

ad

(kN

)

Microstrain







0

50

100

150

200

250

300

350

400

450

0 5000 10000 15000 20000 25000 30000

Ap

plied

Lo

ad

(kN

)

Microstrain







120

Figure 5-36: Specimen DE-A-1.0 20M horizontal reinforcement strains

Figure 5-37: Specimen DE-A-1.0 average flexural reinforcement strains

0

50

100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500 3000 3500 4000 4500

Ap

plied

Lo

ad

(kN

)

Microstrain

Horizontal Bars, East (Num.)

Horizontal Bars, West (Num.)

Horizontal Bars, East (Exp.)

Horizontal Bars, West (Exp.)

0

50

100

150

200

250

300

350

400

450

0 100 200 300 400 500 600 700 800

Ap

plied

Lo

ad

(kN

)

Microstrain





121



Figure 5-39 illustrates the failure patterns of Specimen DE-A-1.0 at various stages. At peak load,

distinct cracking from the re-entrant corner was apparent. The crack at the branch closest to the

re-entrant corner widened until failure, beyond which it consumed the entirety of the nib. For this

specimen, the numerical and experimental shear displacements at failure were nearly identical

and therefore only one instance of tensile damage at failure was shown. Unlike Specimen DE-A-

0.5, the tensile damage was not as smeared and did not spread to the region below the re-entrant

corner.

0

2

4

6

8

10

12

14

16

18

20


Dis

pla

cem

en

t (m

m)


100 kN

200 kN

300 kN

400 kN

421 kN

122

Fig

ure

5-3

9:

Spec

imen

DE

-A-1

.0 f

ail

ure

patt

ern

s at

(a)

pea

k lo

ad;

(b)

num

eric

al

fail

ure

, and n

um

eric

al

fail

ure

corr

espo

ndin

g t

o

exper

imen

tal

shea

r dis

pla

cem

ent

at

fail

ure

; and (

c) e

xper

imen

tal

fail

ure

(a)

(b)

(c)

123


As revealed in the initial analysis of Specimen DE-B-1.0, changing the concrete mesh size

around the vertical stud heads improved strength and ductility. This modification was upheld in

the Poisson’s ratio study, and the result was near identical behaviour. Figure 5-40 and Table 5-6

summarize the updated findings. As for Specimen DE-B-0.5, the vertical studs yielded

considerably, straining to over 14 000 microstrain at failure. However, the horizontal studs

approached yielding but did not do so. The first set of stirrups from the vertical studs remained

below yielding but contributed far more than those of DE-B-0.5. Flexural strains were small and

remained below 120 microstrain. Reinforcement strains for Specimen DE-B-1.0 are shown in

Figure 5-41 to Figure 5-44. The girder vertical displacements at various load levels are shown in

Figure 5-45.

Table 5-6: Summary of Specimen DE-B-1.0 results from additional studies

Girder

Mesh

Size

(mm)

Dilation

Angle

Damage

Regime

Concrete

Poisson’s

Ratio

Numerical

Load

Capacity

(kN)

Numerical

Shear

Displacement at

Failure (mm)

Failure

Location

(End)

15 50° Tension

Only

0.2 403 8.5 Re-entrant

corner

(Hinge) 0 411 9.4

124

Figure 5-40: Effect of concrete Poisson’s ratio on Specimen DE-B-1.0 (mesh size=15 mm,


Figure 5-41: Specimen DE-B-1.0 vertical stud strains

0

50

100

150

200

250

300

350

400

450

0 2 4 6 8 10 12 14

Ap

plied

Lo

ad

(kN

)


Poisson's Ratio=0

Poisson's Ratio=0.2

Experimental

0

50

100

150

200

250

300

350

400

450

0 5000 10000 15000 20000 25000

Ap

plied

Lo

ad

(kN

)

Microstrain

Vertical Studs, East (Num.)

Vertical Studs, West (Num.)

Vertical Studs, East (Exp.)


125

Figure 5-42: Specimen DE-B-1.0 horizontal stud strains

Figure 5-43: Specimen DE-B-1.0 stirrup strains

0

50

100

150

200

250

300

350

400

450

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Ap

plied

Lo

ad

(kN

)

Microstrain





0

50

100

150

200

250

300

350

400

450

0 500 1000 1500 2000 2500

Ap

plied

Lo

ad

(kN

)

Microstrain





126

Figure 5-44: Specimen DE-B-1.0 average flexural reinforcement strains


different load level. Dotted lines show experimental results

0

50

100

150

200

250

300

350

400

450

0 100 200 300 400 500 600 700

Ap

plied

Lo

ad

(kN

)

Microstrain





0

2

4

6

8

10

12

14

16


Dis

pla

cem

en

t (m

m)


100 kN

200 kN

300 kN

395 kN

127

Figure 5-46 illustrates the failure patterns for Specimen DE-B-1.0. Cracks propagated from the

re-entrant corner and the one closest to the re-entrant corner further expanded. A secondary crack

behind the main damage, directly over the support, began to form just as failure occurred;

beyond failure it rapidly spread and covered much of the nib which is not shown below. As in

Specimen DE-A-1.0, the numerical and experimental shear displacements at failure were almost

identical and therefore only one instance of the damage at failure was shown.

128

Fig

ure

5-4

6:

Spec

imen

DE

-B-1

.0 f

ail

ure

patt

erns

at

(a)

pea

k lo

ad;

(b)

num

eric

al

fail

ure

; (c

) nu

mer

ical

fail

ure

corr

espondin

g t

o

exper

imen

tal

shea

r d

ispla

cem

ent

at

fail

ure

; and (

d)

exper

imen

tal

fail

ure

(a)

(b)

(c)

(d)

129

5.1.2.3 Specimen DE-D*-1.0

Out of all the specimens, Specimen DE-D*-1.0 experienced the most profound change in

behaviour with the decrease in concrete Poisson’s ratio. Figure 5-47 and Table 5-7 highlight the

new results. With the new analysis, the dilation angle was increased to 40° as the original value

provided too little ductility. The result was that the ultimate load increased slightly to 438 kN

while the displacement improved to 14.2 mm, which was judged to be reasonably conservative.

No additional studies of DE-D*-1.0 was therefore performed. The strains of all headed studs

exceeded yielding by a wide margin, as shown in Figure 5-49 to Figure 5-51. The strains of the

inclined studs in the two ends were similar, but surprisingly, the vertical stud at the intact end

strained more than that at the failing end. Overall, the vertical and inclined studs strained to

within the range of 15 000-30 000 microstrain. However, unlike in the cases of all previous

applicable specimens, the horizontal studs yielded much more significantly, ranging

approximately between 10 000-20 000 microstrain. The horizontal studs at the failing end ceased

engagement at failure, while the vertical stud stopped straining at some point before failure.

From Figure 5-52 and Figure 5-53, the stirrups contributed little, while Figure 5-54 shows that

the flexural strains were higher than in previous specimens, ranging from 300-500 microstrain

but nonetheless remained far below yielding. The girder vertical deflections at various load

levels are shown in Figure 5-55.

130

Table 5-7: Summary of Specimen DE-D*-1.0 results from additional studies

Girder

Mesh

Size

(mm)

Dilation

Angle

Damage

Regime

Applied

Displacement

(mm)

Concrete

Poisson’s

Ratio

Numerical

Load

Capacity

(kN)

Numerical Shear

Displacement at

Failure (mm)

Failure

Location

(End)

15

35°

Tension

Only 40

0.2 429 13.6

Re-entrant

corner

(Roller)

0

417 6.1

Re-entrant

corner

(Hinge)

40° 438 14.2

Re-entrant

corner

(Roller)

Figure 5-47: Effect of concrete Poisson’s ratio on Specimen DE-D*-1.0 (mesh size=15 mm,


0

50

100

150

200

250

300

350

400

450

500

0 5 10 15 20 25

Ap

plied

Lo

ad

(kN

)


Poisson's Ratio=0

Poisson's Ratio=0.2

Experimental

131

Figure 5-48 Effect of dilation angle on Specimen DE-D*-1.0 (tension damage only, mesh

size=15 mm, ν=0)

Figure 5-49: Specimen DE-D*-1.0 vertical stud strains

0

50

100

150

200

250

300

350

400

450

0 5 10 15 20 25

Ap

plied

Lo

ad

(kN

)


Dilation Angle=35°

Dilation Angle=40°

Experimental

0

50

100

150

200

250

300

350

400

450

0 5000 10000 15000 20000 25000 30000 35000

Ap

plied

Lo

ad

(kN

)

Microstrain

Vertical Stud, East (Num.)

Vertical Stud, West (Num.)

Vertical Stud, East (Exp.)

Vertical Stud, West (Exp.)

132

Figure 5-50: Specimen DE-D*-1.0 inclined stud strains

Figure 5-51: Specimen DE-D*-1.0 horizontal stud strains

0

50

100

150

200

250

300

350

400

450

0 5000 10000 15000 20000 25000 30000

Ap

plied

Lo

ad

(kN

)

Microstrain

Inclined Stud, East (Num.)

Inclined Stud, West (Num.)

Inclined Stud, East (Exp.)

Inclined Stud, West (Exp.)

0

50

100

150

200

250

300

350

400

450

0 5000 10000 15000 20000 25000

Ap

plied

Lo

ad

(kN

)

Microstrain





133

Figure 5-52: Specimen DE-D*-1.0 stirrup strains at east end

Figure 5-53: Specimen DE-D*-1.0 stirrup strains at west end

0

50

100

150

200

250

300

350

400

450

0 200 400 600 800 1000 1200 1400

Ap

plied

Lo

ad

(kN

)

Microstrain





0

50

100

150

200

250

300

350

400

450

0 200 400 600 800 1000 1200

Ap

plied

Lo

ad

(kN

)

Microstrain





134

Figure 5-54: Specimen DE-D*-1.0 average flexural reinforcement strains

Figure 5-55: Specimen DE-D*-1.0 vertical deflection at ends and quarter points of full depth at

different load level. Dotted lines show experimental results

Figure 5-56 shows the failure patterns experienced by Specimen DE-D*-1.0. As with Specimen

DE-D-1.0, multiple cracks propagated from the re-entrant corner; however, as shown in Figure

0

50

100

150

200

250

300

350

400

450

0 100 200 300 400 500 600 700 800 900

Ap

plied

Lo

ad

(kN

)

Microstrain





0

2

4

6

8

10

12

14

16

18

20


Dis

pla

cem

en

t (m

m)


100 kN

200 kN

300 kN

400 kN

413 kN

135

5-56b, at numerical failure, the re-entrant corner cracks worsened to a lesser extent than in

Specimen DE-D-1.0. Instead, a new crack formed near the top of the girder along the nib and

eventually contributed to the failure of the specimen, but from Figure 5-56d the girder top

remained intact. At the point of the analysis corresponding to experimental failure, the severity

of cracking was such that the entire dapped end was smeared with tensile damage, as shown in

Figure 5-56c.

136

Fig

ure

5-5

6:

Sp

ecim

en D

E-D

*-1

.0 f

ail

ure

patt

erns

at

(a)

pea

k lo

ad;

(b)

num

eric

al

fail

ure

; (c

) num

eric

al

fail

ure

corr

espond

ing t

o

exper

imen

tal

shea

r dis

pla

cem

ent

at

fail

ure

; and (

d)

exper

imen

tal

fail

ure

(a)

(b)

(c)

(d)

137

5.1.3 Energy Check

As mentioned in the previous chapter, the primary check to verify that an analysis performed in

the explicit solver is quasi-static, is to ensure that the ratio of kinetic energy to internal energy of

all deformable bodies is small, recommended to be 5% at most. At the start of a quasi-static

analysis, this is generally difficult to achieve because movement of the components is already

present even before primary deformation takes place. Because, in absolute terms, the values of

the energies are small at the start, a small change in one type of energy may represent a

significant change relative to the other type of energy. In all seven specimens analyzed in this

study, this was the case indeed. Figure 5-57 shows the kinetic to internal energy ratio for all

seven specimens. Kinetic energy was generally 200-600% of the internal energy, in the extreme

case of Specimen DE-D*-1.0 the ratio reached a peak value of close to 2 600%. However, these

extremes occurred not more than 10% into the analyses in all cases, which corroborates the idea

that the energy ratio typically cannot be satisfied early on. In addition, the specimens

experienced smaller secondary peaks in kinetic energy within the range of 10-25% analysis

completion, albeit at a significantly lower magnitude than at the beginning and generally

remaining below 5% of internal energy. These correspond to initial crack formation which may

induce sudden changes in the movement of the supports. The effect was most pronounced in

Specimen DE-B-1.0 where the kinetic to internal energy ratio reached a maximum value of

11.8% following the default peaks experienced by all the specimens. Beyond 25% analyses

completion, kinetic energy remained well below 1% of internal energy in all specimens. Small

peaks were typically experienced near the points in the analyses when failure occurred.

However, significant deformations have already taken place up to failure which corresponded to

a large internal energy; therefore kinetic energy became negligible even though failure occurred.

138

Figure 5-57: Kinetic to internal energy ratio for all specimens, taken from the best cases in the

additional (final) studies (mesh size=15 mm, tension damage only, ν=0.2). Values

beyond 12% are not shown.

5.1.4 Effect of Stud Head Bearing Concrete

One of the advantages of using headed studs is that the heads provide anchorage as a result of

bearing action against the concrete directly adjacent to the heads in the direction of the stem.

Thus, the concrete forming part of the compressive strut is used effectively as its strength is

engaged. Figure 5-58 to Figure 5-63 show the distribution of concrete stresses within the vicinity

of the stud heads due to bearing action in Specimens DE-B-0.5, DE-C-1.0, and DE-D-1.0 at their

respective numerical failure points. Each figure shows a cut section corresponding to the stud

indicated. The dark blue regions are those where the compressive capacities of the concrete have

either been reached or exceeded, indicating that the studs are providing effective anchorage

through bearing on the concrete.

0

2

4

6

8

10

12

0 10 20 30 40 50 60 70 80 90 100

Rati

o o

f K

ineti

c t

o In

tern

al

En

erg

y o

f M

od

el

(%)

Analysis Completion (%)

DE-A-0.5

DE-A-1.0

DE-B-0.5

DE-B-1.0

DE-C-1.0

DE-D-1.0

DE-D*-1.0

Up to 2597%

139

Figure 5-58: Concrete bearing stress adjacent to horizontal studs in Specimen DE-B-0.5. Mesh

size=15 mm, dilation angle=55°, tension damage only, concrete Poisson’s ratio=0

Figure 5-59: Concrete bearing stress adjacent to vertical studs in Specimen DE-B-0.5. Mesh


140

Figure 5-60: Concrete bearing stress adjacent to inclined studs in Specimen DE-C-1.0. Mesh




141

Figure 5-62: Concrete bearing stress adjacent to inclined studs in Specimen DE-D-1.0. Mesh




142

5.2 Summary of Primary Analyses

The following model sensitivity parameters were chosen for final consideration of the specimen

behaviour:

Average concrete girder body mesh size: 15 mm

Damage regime: Only tensile damage included; concrete in compression assumed intact

Dilation angle: 30° to 40° for specimens with diagonal reinforcement; 50° to 55°

otherwise.

Applied displacement for Specimens DE-D/D*-1.0 increased to 40 mm from 25 mm.

Table 5-8 summarizes the results for the best model cases produced by the parameters mentioned

above.

Table 5-8: Summary of specimen results for final cases selected

Specimen Load Capacity (kN) Shear Displacement

at Failure (mm) Failure Location

DE-A-0.5 471 15.3 Re-entrant corner

DE-A-1.0 403 10.3 Re-entrant corner

DE-B-0.5 427 10.2 Re-entrant corner

DE-B-1.0 411 9.4 Re-entrant corner

DE-C-1.0 547 4.4 Full depth beside re-

entrant corner

DE-D-1.0 418 17.5 Re-entrant corner

DE-D*-1.0 438 14.2 Re-entrant corner

The behaviour of the specimens and the effect each studied parameter had on them will be

discussed in Chapter 6.

143

5.3 Additional Refinements to Concrete Girder Mesh

Despite strength capacities and shear ductilities being well-captured by the finite element models

with the selected final parameters, further refinement was necessary to eliminate the anomalies in

the elastic region of the load-displacement behaviour of some of the specimens whereby the

curves are shifted with respect to the experimental data. This effect was most noticeable in

Specimens DE-B-0.5/1.0 and DE-D/D*-1.0. The stud cavities within the concrete girder, as well

as the headed studs themselves, were therefore prescribed a finer mesh size. Two specimens,

DE-B-0.5 and DE-D-1.0, were chosen for the refinement analyses and the results are presented

below in Tables Table 5-9 and Table 5-10 and Figure 5-64-Figure 5-66.

Table 5-9: Summary of Specimen DE-B-0.5 results from mesh refinement

Average

Number

of

Elements

Around

Circular

Openings

Dilation

Angle

Damage

Regime

Concrete

Poisson’s

Ratio

Numerical

Load

Capacity

(kN)

Numerical Shear

Displacement at

Failure (mm)

Failure

Location

(End)

8

55°

Tension

Only

0

403 10.3

Re-entrant

corner

(Hinge)

22 442 13.2

Re-entrant

corner

(Hinge)

22 50° 435 11.8

Re-entrant

corner

(Roller)

144

Figure 5-64: Effect of mesh refinement on the behaviour of Specimen DE-B-0.5. Tension damage

only, concrete Poisson’s ratio=0.

For Specimen DE-B-0.5, refining the mesh around the circular cavities by roughly three times

significantly reduces the instability experienced in the ascending branch of the load-displacement

progression. Using the previously chosen dilation angle of 55°, the strength and capacity are

over-predicted at 442 kN and 13.2 mm, respectively. As a result, the dilation angle was reduced

to 50°, yielding a strength of 435 kN and shear displacement of 11.8 mm at failure.

0

50

100

150

200

250

300

350

400

450

500

0 2 4 6 8 10 12 14

Ap

plied

Lo

ad

(kN

)


Unrefined-Dilation Angle=55°

Refined-Dilation Angle=55°

Refined: Dilation Angle=50°

Experimental

145

Table 5-10: Summary of Specimen DE-D-1.0 results from mesh refinement

Average

Number

of

Elements

Around

Circular

Openings

Dilation

Angle

Damage

Regime

Concrete

Poisson’s

Ratio

Applied

Displace-

ment

(mm)

Numerical

Load

Capacity

(kN)

Numerical

Shear

Displacement

at Failure

(mm)

Failure

Location

(End)

8

35°

Tension

Only

0

40

418 17.5

Re-

entrant

corner

(Hinge)

22 480

Did not

completely

fail

Full

depth

(Hinge)

22 30° 436 14.3

Full

depth

(Hinge)

22 40° 482

Did not

completely

fail

Full

depth

(Hinge)

22 40° 60 483 27.7

Full

depth

(Hinge)

146

Figure 5-65: Effect of mesh refinement on the behaviour of Specimen DE-D-1.0. Tension damage

only, concrete Poisson’s ratio=0.

Figure 5-66: Effect of increased applied displacement from 40 mm to 60 mm on the behaviour of

Specimen DE-D-1.0. Tension damage only, concrete Poisson’s ratio=0.

0

50

100

150

200

250

300

350

400

450

500

550

0 5 10 15 20 25

Ap

plied

Lo

ad

(kN

)


Unrefined - Dilation Angle=35°

Refined - Dilation Angle=35°

Refined - Dilation Angle=30°

Refined-Dilation Angle=40°

Experimental

0

50

100

150

200

250

300

350

400

450

500

550

0 5 10 15 20 25 30 35

Ap

plied

Lo

ad

(kN

)


Experimental

Refined-DilationAngle=40°, Disp.=40mm

Refined-DilationAngle=40°, Disp.=60 mm

147

Refining the mesh around stud cavities for Specimen DE-D-1.0 did not yield results which were

significantly improved over the unrefined results. The instability in the elastic region causing the

shift of load-displacement is clearly eliminated, but in the non-linear region small fluctuations

were unavoidable. It was found that for the same value of dilation angle used under a refined

mesh, the strength increased to around 480 kN and did not fail at any point in the analysis.

Decreasing the dilation angle to 30° revealed that failure occurs in the full depth portion of the

girder, with the strength being 436 kN and shear displacement at failure being roughly 14 mm.

Previous analysis of Specimen DE-D-1.0 showed that the failure location may be redirected back

to the re-entrant corner through an increase in dilation angle, but as in the 35° situation, the

specimen did not fail. As a result, the applied displacement was increased further from 40 mm to

60 mm, and the result was a similar strength but shear displacement of 27 mm at failure.

148

CHAPTER 6 - DISCUSSION

6.1 General

The impact that the investigated parameters of mesh size, dilation angle, and type of damage had

on the structural behaviour varied from specimen to specimen. Some specimens were more

sensitive to variations in certain parameters than others were. Of course, the sensitivities and

their implications present in this study may not apply to all types of structures modelled; the

choice of parameters selected may very well have different effects on slabs, columns,

connections, corbels, and beams with varying significance. In this chapter the effect of the

various factors examined will be discussed.

6.2 Effect of Mesh Size

With regards to mesh size, in general the elastic behaviour of the specimens was nearly identical.

Differences in mesh sizes became apparent only when the specimens either reached the peak

load or entered the inelastic region. Beyond failure, the correlation between the mesh sizes

deteriorated further. While a stress-to-crack width displacement relation was initially specified to

minimize the effects of mesh size, the sensitivity remained because, in the CDP model, a

characteristic crack length is present across an element if a stress-to-crack displacement

relationship, coupled with fracture energy, is considered. This length is introduced such that the

direction of cracking is objective and can occur in any direction, as it is unknown beforehand. As

a result, it is recommended to use element sizes with an aspect ratio close to one, which is

measured as the ratio of the crack width to element size. Smaller elements, for the same crack

width, display larger aspect ratios and therefore sensitivity cannot be eliminated. Within the

current studies, it was difficult to satisfy the aspect ratio recommendation. Especially for the

149

specimens where numerous diagonal partitions were present due to the inclined studs, hexahedral

element shapes that were not cubic or rectangular were unavoidable. As well, large strains in the

principal directions perpendicular to the crack directions further deform the elements such that

the characteristic lengths become increasingly erroneous. Smaller elements using the CDP model

therefore do not necessary yield convergent behaviour. While the 15 mm mesh size was

considered to be fairly coarse, it produced reasonable behaviour. Of course, the mesh should not

be overly course and balance must be achieved. The aforementioned explanations are supported

by Specimens DE-C-1.0 and the mesh cases for DE-D-1.0 where failure occurred in the full

depth portion of the girder. In said cases, there was little difference in the load-displacement

behaviour between the mesh sizes because the failure that took place in the full depth did not

result in elements as deformed as those which failed at the re-entrant corners. In addition, the

element shapes immediately adjacent to the full depth were more rectangular than those at the re-

entrant corners. In terms of failure patterns, the smaller mesh sizes produced a smaller area over

which tensile damage took place, whereas the damage for the larger meshes was more smeared.

6.3 Effect of Dilation Angle

In many cases where failure occurred at the dapped ends, the load did not drop to zero, but rather

decreased to some value 40% to 60% of that just prior to failure. The load either remained

relatively constant at this level or experienced a second decrease indicative of another failure

occurring after the primary one. The presence of this residual strength may be attributed to the

combination of the presence of concrete elements, even after full damage is achieved, and its

continued interaction with steel after failure. In the CDP model, elements do not get removed

even after cracking takes place. This allowed the reinforcement to continue to engage so long as

some shear interaction with the concrete remained intact. It can be seen in Figure 6-1 that some

150

of the reinforcement strains, most noticeably in the horizontal studs of Specimens DE-B-0.5/1.0

and DE-D-1.0, snapped back at or near failure, and then continued to increase in strain at some

point afterwards.

Figure 6-1: “Snap-back” behaviour present in east horizontal stud of Specimen DE-B-1.0.

Results for trial simulation shown, not used for sensitivity analyses.

The “staircase” load-displacement behaviour resulting from this action during the post-failure

state was more susceptible in the more brittle cases, namely the analyses with lower dilation

angles. For higher values, as the ductility generally increased, the phenomenon became less

pronounced and hence it was more difficult to determine the point of failure. In the D specimens,

for example, the sudden drop in load and increase in shear displacement was more subtle.

However, for all the specimens, a strong indication of failure was a sudden increase in the

amount of tensile damage experienced at the failing end; typically at the point of the analysis just

beyond failure, tensile damage quickly and suddenly engulfed the nib of the girder. The

behaviour of the specimens beyond failure was therefore neglected and the residual strength

151

present was judged ineffective. The general increase in strength and ductility with greater values

of dilation angle is explained by recalling that in geomaterials the dilation angle is a measure of

volumetric change due to shear deformations. The greater the dilation angle, the greater the force

each individual granule must overcome in order to move past adjacent granules, and hence the

greater the volumetric strain. It was found that all the specimens with either identical or similar

reinforcing layouts shared similar values of dilation angle. The dilation angle can be affected by

the degree of confinement experienced by the concrete. From a broad sense, Specimens DE-A-

0.5/1.0 had the highest confinement at the re-entrant corner with the presence of the two closely

spaced stirrups, Specimen DE-C-1.0 would have a relatively intermediate degree of confinement,

while Specimens DE-D/D*-1.0 and DE-B-0.5/1.0 would possess the lowest confinement as none

of the concrete at or near the re-entrant corners in these specimens were enclosed by stirrups.

However, there was no definite correlation between the confinement level and selected dilation

angle. Lower dilation angles would be expected for the B and D specimens since, as mentioned,

they lack stirrups at the re-entrant corner. A high value of 55° and 50° was nonetheless specified

for the Specimens DE-B-0.5 and DE-B-1.0, respectively, but on the other hand, lower dilation

angles such as 35° and 40° were sufficient in predicting the behaviour of Specimens DE-D-1.0

and DE-D*-1.0, respectively. However, the behaviour of the A specimens was predicted using

relatively high dilation angles of 50° or more; these specimens had the highest level of

confinement. Specimen DE-C-1.0 required the lowest dilation angle value of 30°, but failure

took place away from the re-entrant corner in the full depth which would have different

implications for the appropriate dilation angle to use. However, it appears that the presence of

diagonal reinforcement affects the dilation angle value to be used. In the C and D specimens

where diagonal studs are used the dilation angles assigned did not exceed 40° while for the A

152

and B specimens very high values equaling 50° or greater were needed to match experimental

behaviour in terms of strength and ductility. In summary, the dilation angle analysis has shown

that values chosen for specimens whose predicted failure is at the full girder depth may be lower

than those which predict failure at the dapped ends, but it is in general difficult to predict exactly

which value would be appropriate and hence dilation angle sensitivity analyses must be

performed using the CDP material model.

6.4 Effect of Concrete Damage

The behaviour of the specimens was somewhat dependent upon whether both compressive and

tensile damage was included, or if tensile damage only was considered. In all the reference

specimens where the analyses were conducted, the exclusion of compressive damage had little

impact on the predicted ultimate strength. For Specimens DE-A-0.5, DE-B-0.5, and DE-C-1.0

the exclusion of compression damage had almost no effect on load-displacement behaviour in

the pre-failure state. For Specimen DE-D-1.0 the lack of compression damage resulted in the

absence of girder failure altogether under the default applied loading. Lastly, in all cases where

neither compressive nor tensile damage was included the ultimate loads were overestimated, and

in some situations by an excessive amount; these cases were checked but immediately discarded.

When no damage was present, shear displacements at failure were generally difficult to pinpoint

as the cracking and loads progressed very gradually, without any clear decreases in load after

failure. In the experiments, it was indeed true that tensile cracking from shear failure would

govern specimen behaviour to a far greater extent than concrete crushing. However, in the full

damage models it was found that substantial compressive damage took place at roughly the same

locations as the tensile damage; both types of damage were being experienced at the failure

locations. However, a possible reason for this occurrence may be explained by the manner in

153

which an individual element deformed. As shown in Figure 6-2, a truss mechanism was proposed

to explain the simultaneous presence of compression and tensile damage. If a typical element is

loaded in tension in an arbitrary direction, it will experience compression in the perpendicular

direction due to truss action, similar to how tension acting on ties arranged in a diamond

configuration will create compression in the strut placed perpendicularly, in the middle, to the

applied tension forces. The result is that the element experiences tension in one direction and

compression in the orthogonal direction, thus accounting for the presence of both types of

damage. The final decision to consider only tensile damage implied the assumption that cracking

failure of concrete would occur much before crushing failure, and crushing failure would not

take place at all throughout the analyses.

Figure 6-2: Possible source of compression damage at crack location

6.5 Effect of Mesh Refinement around Cavities Created by Reinforcement

The discontinuities near the beginning happen at the onset of the first crack at the re-entrant

corner. The contact between the steel and concrete is not perfect and may disengage and then

154

reengage as the concrete elements shift and deform, causing the increasing and decreasing

behaviour. Lastly, because the loading is displacement controlled, load-displacement behaviour

is captured in more detail without causing singularities, meaning the effect of material damage

and boundary conditions is more obvious. One previous analysis with Specimen DE-D-1.0 using

load control was performed and the result was smooth behaviour. A finer mesh around the holes

significantly reduced the initial discontinuity; however, small fluctuations beyond the elastic

range persisted. As the concrete is experiencing damage it becomes increasingly difficult for

uniform contact behaviour to be maintained between itself and the reinforcement; contact

properties are most applicable when the interface materials are within their linear regions. For

Specimen DE-D-1.0, the refined mesh resulted in improved contact between the studs and

concrete, which was enough to prevent failure from occurring at the re-entrant corner. This is

contrary to the experiment and it is possible that the refined mesh has yet further impact on the

appropriate material and contact parameters used.

6.6 Implications for Shear Friction and Diagonal Bending Methods

From the analytical methods carried out by Herzinger and El-Badry (2007), it was found that

shear friction provided accurate predictions for the strength of the Specimens DE-A-0.5/1.0 and

DE-B-0.5/1.0 specimens, but overestimated the strengths of Specimens DE-C-1.0 and DE-D/D*-

1.0. Conversely, diagonal bending accurately predicted the load capacities of the majority of the

models; even for Specimen DE-B-1.0 the magnitude in which the strength was overestimated

was lower than that of the shear friction results for the specimens in which the larger errors were

present. Results shown by the finite element models reinforce the premise that shear friction is

not conservative enough in determining the load capacities of the specimens where diagonal

reinforcement is present. Even though the numerical models for said specimens predicted a load

155

slightly higher than those determined experimentally, in all cases they remained below the values

given by the shear friction method. That being said, results from the numerical models imply that

the diagonal bending method is the more conservative approach especially in the design of

members with diagonal shear reinforcement. The diagonal bending method is more accurate for

specimens with diagonal reinforcement because the horizontal component of the axial forces in

the diagonal reinforcement is accounted for by the compression in the concrete region above the

neutral axis. In the shear friction method the section is fully cracked along the girder depth

meaning the compressive capacity of the concrete is not utilized. However, shear friction

generally appears to be more suitable in the design of members without diagonal reinforcement;

it provided more accurate strength predictions than diagonal bending in the analysis of the B

specimens, for example. Coupled with experimental behaviour, the finite element results show

that both shear friction and diagonal bending accurately predicted the failure locations and the

approximate orientations of the cracks.

156

CHAPTER 7 - SUMMARY, CONCLUSIONS, & FUTURE WORK

7.1 Summary

A numerical study was conducted to model the behaviour of seven dapped-ended concrete

girders reinforced with headed studs and tested by Herzinger and El-Badry (2007). The finite

element software ABAQUS was used in a three dimensional, non-linear analysis. Sensitivity

analyses were performed on the mesh size, concrete dilation angle, type of damage, and finally

Poisson’s ratio of concrete. The load versus 45° shear displacement behaviour was the primary

result sought out in each model. In addition, the reinforcement strains, the vertical displacements

at quarter points on the girders at various load levels, and damage patterns, were checked. The

final results not only attempted to validate those from the experiments, but also check the

accuracy of the shear friction and diagonal bending analysis methods.

7.2 Conclusions

Overall, the behaviour of the specimens was well predicted by the initially developed finite

element models. Careful selection of mesh size, dilation angle, and damage type is necessary for

accurate modelling of dapped-ended girders reinforced with headed studs. The primary

conclusions initially drawn from the current research are:

A concrete girder mesh size of 15 mm, despite being somewhat coarse, was adequate in

modelling the behaviour of the specimens. Finer mesh sizes in the CDP model do not

necessarily correspond to better results or more convergent behaviour. A mesh size of at

least maximum concrete aggregate size should be chosen, but not resulting in an overly

coarse mesh.

157

Dilation angles in general encompassed a wide range of values which varied from 30° to

55°. This parameter had significant effects on the strength and ductility of the specimens,

and in some situations the failure locations. There was little correlation between the

appropriate value to use based on the concrete material property; however, the

reinforcing layout appears to have a significant effect on the appropriate values to be

chosen. It is therefore recommended that, for specimens where inclined reinforcement is

used, lower values of dilation angle may be assumed, otherwise very high values in

excess of 50° may be required to compensate for the extremely brittle behaviour

inherently present at the girder dapped ends.

In the analysis of dapped-ended girders using the CDP model, it is acceptable to exclude

compression damage as much of the failure will be tensile in nature. The exclusion of

damage altogether should be avoided as it may overestimate the strength capacities for

the same material parameters used. The points in time during the analyses corresponding

to failure must be interpreted carefully using indications such as load drops and sudden

increase in damage.

Both the numerical and experimental programs show that diagonal bending gives the

best prediction of the strength of the specimens where inclined studs were present. On

the other hand, shear friction may adequately predict the maximum loads of specimens

where only vertical and horizontal shear reinforcement is used. As a result, it is

recommended that the diagonal bending method should be used for any design involving

inclined reinforcement orientations, while shear friction may be used where strictly

vertical and horizontal reinforcement are present.

158

While the initial models predicted the behaviour of the specimens with reasonable accuracy,

mesh refinements were made to reduce load-displacement instability. The following conclusions

are drawn from the additional simulations performed implementing the changes:

Refining the mesh around the circular stud cavities within the concrete girder reduces

instabilities which cause the load-displacement behaviour to shift

Minor fluctuations may not be eliminated completely beyond the elastic range of the

specimens and are likely a result of changing contact behaviour due to damage occurring

in the concrete material

With refinement in the mesh, specimens with horizontal, vertical, and inclined layouts

may show failure in the full depth portion of the girder as opposed to re-entrant corners.

7.3 Recommendations for Future Work

By establishing the current numerical models which have been shown to accurately predict the

behaviour of dapped-ended girders with headed studs, parametric studies should be performed in

the future. Girder dimensions, stud layouts, loading, and support conditions can be investigated

entirely using finite element analysis. Behaviour of the specimens under dynamic loading should

be explored to examine the effect that moving loads will have, especially since dapped-ended

girders are widely used in bridge structures Lastly, mesh refinement around contact regions

should be investigated in greater detail in future studies, and the resulting effects on the

appropriate finite element parameters to be used should also be examined. Analysis parameters

relevant to quasi-static loading may also be explored in order to optimize the efficiency of the

computations.

159

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162

Appendix I – E-mail Correspondence Regarding Permission to Use

Content

RE: Permission to Use Data Mamdouh El-Badry <[email protected]> Wed 2020-01-29 10:20 PM To: Kevin Wing-Chung Yuen <[email protected]>

Dear Mr. Kevin Yuen, As your MSc research is extension of previous research done by my former PhD student under my supervision, and since it is necessary to compare the results of your numerical analyses to those obtained from the previous experimental tests and analytical studies, I hereby confirm that you have my permission to use in your MSc thesis Figures 4-6, 9, and 10, and Table 1, and reproduce Figures 7 and 8, from the paper by Herzinger, R. and El-Badry, M., “Alternative Reinforcing Details in Dapped Ends of Precast Concrete Bridge Girders,” Transportation Research Record Journal of the Transportation Research Board, 2013. However, please note that your use of these data must be limited to your thesis only. Sincerely, Dr. Mamdouh El-Badry --------------------------------------------------

Mamdouh El-Badry, PhD, PEng, FCSCE Professor of Civil Engineering University of Calgary 2500 University Drive, N.W. Calgary, Alberta, Canada T2N 1N4 Tel: (403) 220-5819 Fax: (403) 282-7026 E-mail: [email protected]

From: Kevin Wing-Chung Yuen <[email protected]> Sent: January 29, 2020 9:30 PM To: Mamdouh El-Badry <[email protected]> Subject: Permission to Use Data

Dear Dr. El-Badry I am writing this email to use the experimental data on previous research regarding "Alternative Reinforcing Details in Dapped Ends of Precast Concrete Bridge Girders" for literature review and comparison with my numerical studies. With your permission, I would like to use Figures 4-6, 9, and 10, and Table 1, and reproduce Figures 7 and 8, from the aforementioned paper, in my thesis titled "Finite Element Analysis of Dapped-Ended Concrete Girders Reinforced with Steel Headed Studs" to be submitted to University of Calgary Faculty of Graduate Studies. Thank you for the consideration Regards ______________________________________________ Kevin Yuen, EIT MSc Student Department of Civil Engineering

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Schulich School of Engineering University of Calgary E-mail: [email protected] Office: ENF 319A Address: 2500 University Drive NW, Calgary AB, T2N 1N4

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