This thesis deals with the behaviour of composite steel-concrete beams with partial shear
connexion. The goal of this study is to develop and implement numerical tools which
are able to predict the short and long-term behaviour of composite steel-concrete beams.
The first part concerns the modelling of composite beams in the linear elastic range
in which two bond models at the interface are considered : discrete bond and distributed
bond. A finite element with exact stiffness matrix is developed in order to conduct a
critical analysis of these two bond models. In the second part, the time-dependent
behaviour of the concrete (creep and shrinkage) is considered by adopting a linear viscoelastic
model. An original semi-analytical solution is proposed for the two bond models.
This solution enables the analysis of the time-dependent behaviour of composite beams
and to evaluate the performances of simplified viscoelastic approaches for concrete creep.
The third part deals with the constitutive modelling of the materials (steel, concrete
and connector) based on nonlinear continuum mechanics concepts. A coupled elastoplastic
damage model for concrete is proposed. The fourth part is dedicated to the development
of three nonlinear F.E. formulations (displacement-based, force-based and two-field
mixed formulation) for composite beams and for the two bond models. An original state
determination, taking into account the element internal load, is proposed for the forcebased
and two-field mixed formulations. Finally, in the last part, we propose, as a
first approach, a viscoelastic/plastic model for concrete in order to simulate the interaction
between the time-dependent effects and the cracking of concrete.
Modelling of the non-linear behaviour of composite beams
1. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Modelling of the non-linear behaviour of
composite beams
taking into account the time effects
Quang-Huy NGUYEN
INSA de Rennes - Structural Engineering Research Group
University of Wollongong - Faculty of Engineering
13 July 2009
Q-H. Nguyen PhD Thesis Defense
6. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Outline
1 Introduction
2 Elastic analysis of composite beams
3 Time-Dependent Behaviour
4 Nonlinear Behaviour of Materials
5 Finite Element Formulations
6 Time-Dependent Behaviour In the Plastic Range
Q-H. Nguyen PhD Thesis Defense
7. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Outline
1 Introduction
2 Elastic analysis of composite beams
3 Time-Dependent Behaviour
4 Nonlinear Behaviour of Materials
5 Finite Element Formulations
6 Time-Dependent Behaviour In the Plastic Range
7 Conclusions and Futur works
Q-H. Nguyen PhD Thesis Defense
8. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Outline
1 Introduction
2 Elastic analysis of composite beams
3 Time-Dependent Behaviour
4 Nonlinear Behaviour of Materials
5 Finite Element Formulations
6 Time-Dependent Behaviour In the Plastic Range
7 Conclusions and Futur works
Q-H. Nguyen PhD Thesis Defense
9. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Outline
1 Introduction
General
Background: Analysis of composite beams
Research questions
Objectives
Q-H. Nguyen PhD Thesis Defense
10. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
General
Introduction to steel-concrete composite beam
Steel-concrete composite structure are widely used in the construction
industry
Economic
Reduced live load deflections
Reduced weight
Fast erection process
Increased span lengths are possible
Stiffer floors
Composite beam system (Ricker 1989)
Q-H. Nguyen PhD Thesis Defense
11. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
General
Introduction to steel-concrete composite beam
Steel-concrete composite structure are widely used in the construction
industry
Economic
Reduced live load deflections
Reduced weight
Fast erection process
Increased span lengths are possible
Stiffer floors
Composite beam system (Ricker 1989)
Q-H. Nguyen PhD Thesis Defense
12. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
General
Introduction to steel-concrete composite beam
Composite beams consist of steel beam and concrete slab joint
together as a unit by shear studs
steel beam
shear stud
concrete slab
profile sheeting
reinforcement
Q-H. Nguyen PhD Thesis Defense
13. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Bond models
A
A
B
B
section A-A section B-B
Q-H. Nguyen PhD Thesis Defense
14. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Bond models
A
A
B
B
section A-A section B-B
Discrete bond model
Aribert (1982, France)
Schanzenback (1988, Germany)
Q-H. Nguyen PhD Thesis Defense
15. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Bond models
A
A
B
B
section A-A section B-B
Discrete bond model
Aribert (1982, France)
Schanzenback (1988, Germany)
Distributed bond model
Newmark (1951, US)
Adekola (1968, Nigeria)
Q-H. Nguyen PhD Thesis Defense
16. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Analysis Type
Elastic Analysis Inelastic Analysis
Newmark, 1951
Adekola, 1968
N
N
tM
x
scd
X
Y
2
2
12
d ( )
( ) ( )
d
t
N x
N x C M x
x
μ− = ⇒ Analytical solution
Q-H. Nguyen PhD Thesis Defense
17. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Inelastic Analysis
Displacement-based Force-based Mixed
1v
2v
2u
1u
θ1
θ2
x
( )v x
( )u x
X
Y
θ
θ
⎡ ⎤
⎢ ⎥
⎢ ⎥
⎡ ⎤ ⎢ ⎥
=⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥⎣ ⎦
⎢ ⎥
⎢ ⎥
⎢ ⎥⎣ ⎦
1
1
1
2
2
2
( )
( )
( )
u
v
u x
x uv x
v
a
Assumed displacement field
Q-H. Nguyen PhD Thesis Defense
19. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Inelastic Analysis
Displacement-based Force-based Mixed
⎡ ⎤
⎡ ⎤ ⎢ ⎥
=⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥⎣ ⎦
⎢ ⎥⎣ ⎦
1
2
2
( )
( )
( )
M
N x
x M
M x
N
b
X
Y
1M
1M
2N
( )N x
( )M x
x
Assumed force field
Q-H. Nguyen PhD Thesis Defense
23. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Inelastic Analysis
Concentrated Plasticity Distributed Plasticity
-endi -endj
Inelasticity is lumped at member ends
elastic member
Q-H. Nguyen PhD Thesis Defense
24. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Inelastic Analysis
Concentrated Plasticity Distributed Plasticity
The element behavior is
monitored along its length
-endi -endj
Fiber element model
Fiber section
Q-H. Nguyen PhD Thesis Defense
25. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Inelastic Analysis
Concentrated Plasticity Distributed Plasticity
Fiber section model
εc
σc
σs
εs
Concrete fiber
Steel fiber
σ σ
σ σ
=
=
=
=
∑∫
∑∫
1
1
d
d
n
i i
iA
n
y i i i
iA
N A A
M z A A z
Fiber discretization
of cross-section
y
z
Arizumi et al., 1981
Fiber element model
Cross-section behavior
Q-H. Nguyen PhD Thesis Defense
26. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Inelastic Analysis
Concentrated Plasticity Distributed Plasticity
Fiber element model
Cross-section behavior
Fiber section model
Macro model
El-Tawil and Deierlein, 2001
Bounding Surface
Axial Force
Moment
Loading Surface
Compression Region
Tension Region
Stress-resultant Plasticity Models
Q-H. Nguyen PhD Thesis Defense
27. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Analysis Type
Elastic Analysis
Time effects
Inelastic Analysis
Time Effects
Gilbert, 1989
Boerave, 1991
Amadio and Fragiacomo, 1993
Dezi and Tarantino, 1993
...
Q-H. Nguyen PhD Thesis Defense
28. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Background: Analysis of composite beams
Analysis Type
Elastic Analysis
Time effects
Inelastic Analysis
Time Effects
Gilbert, 1989
Boerave, 1991
Amadio and Fragiacomo, 1993
Dezi and Tarantino, 1993
...
Q-H. Nguyen PhD Thesis Defense
29. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Research questions
1 Discrete bond model or distributed bond model?
2 Displacement-based, Force-based or Mixed formulation?
3 What is the influence of creep and shrinkage on the
behaviour of composite beams?
4 How to take into account the time effects in inelastic
analysis?
Q-H. Nguyen PhD Thesis Defense
30. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Objectives
The main objectives are:
1 Discrete versus distributed bond modelling
2 To study the time effects in composite beams
(viscoelastic model)
3 To develop three non-linear F.E. formulations and to
study their performances for both bond models
4 To combine time effects and cracking of concrete
Q-H. Nguyen PhD Thesis Defense
31. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Outline
2 Elastic analysis of composite beams
Basic assumptions
Governing Equations of Composite Steel-Concrete Beams
Exact Stiffness Matrix - Elastic behaviour
Comparison of the two bond models
Q-H. Nguyen PhD Thesis Defense
32. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Basic assumptions
1 Euler-Bernoulli’s assumption for both the slab and the profile
2 Slip can occur at the slab/profile interface but no uplift
3 Deformations and displacements remain small
4 Local buckling and torsional stress are not accounted for
5 Fiber discretization to describe section behaviour
6 Spring model to describe the force transfer mechanism through
bond
Q-H. Nguyen PhD Thesis Defense
33. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Basic assumptions
1 Euler-Bernoulli’s assumption for both the slab and the profile
2 Slip can occur at the slab/profile interface but no uplift
3 Deformations and displacements remain small
4 Local buckling and torsional stress are not accounted for
5 Fiber discretization to describe section behaviour
6 Spring model to describe the force transfer mechanism through
bond
Q-H. Nguyen PhD Thesis Defense
34. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Basic assumptions
1 Euler-Bernoulli’s assumption for both the slab and the profile
2 Slip can occur at the slab/profile interface but no uplift
3 Deformations and displacements remain small
4 Local buckling and torsional stress are not accounted for
5 Fiber discretization to describe section behaviour
6 Spring model to describe the force transfer mechanism through
bond
Q-H. Nguyen PhD Thesis Defense
35. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Basic assumptions
1 Euler-Bernoulli’s assumption for both the slab and the profile
2 Slip can occur at the slab/profile interface but no uplift
3 Deformations and displacements remain small
4 Local buckling and torsional stress are not accounted for
5 Fiber discretization to describe section behaviour
6 Spring model to describe the force transfer mechanism through
bond
Q-H. Nguyen PhD Thesis Defense
36. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Basic assumptions
1 Euler-Bernoulli’s assumption for both the slab and the profile
2 Slip can occur at the slab/profile interface but no uplift
3 Deformations and displacements remain small
4 Local buckling and torsional stress are not accounted for
5 Fiber discretization to describe section behaviour
6 Spring model to describe the force transfer mechanism through
bond
Q-H. Nguyen PhD Thesis Defense
37. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Basic assumptions
1 Euler-Bernoulli’s assumption for both the slab and the profile
2 Slip can occur at the slab/profile interface but no uplift
3 Deformations and displacements remain small
4 Local buckling and torsional stress are not accounted for
5 Fiber discretization to describe section behaviour
6 Spring model to describe the force transfer mechanism through
bond
Q-H. Nguyen PhD Thesis Defense
38. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Governing Equations of Composite Steel-Concrete Beams
1 Equilibrium
Distributed bond
Discrete bond
2 Compatibility
3 Constitutive relations
2
2
d ( )
( ) 0
d
d ( )
( ) 0
d
d ( ) d ( )
0
dd
c
sc
s
sc
sc
z
N x
D x
x
N x
D x
x
M x D x
H p
xx
+ =
− =
+ + =
=sc sc eD∂ − ∂ −D P 0
Matrix form
zp
cH
cM dc cM M+
dc cN N+
dc cT T+
cN
cT
scV
scD
dx
ds sM M+
ds sN N+
ds sT T+
sM
sN
sT
scD
sH
x
z
y
Q-H. Nguyen PhD Thesis Defense
39. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Governing Equations of Composite Steel-Concrete Beams
1 Equilibrium
Distributed bond
Discrete bond
2 Compatibility
3 Constitutive relations
unconnected element
cN +
cM +
sN +
sM +
sN −
sM −
cN −
cM −
cN
cM
sN
sM
stQ
stQ
0xΔ =
connector element
=e∂ −D P 0
Unconnected beam segment Single connector
1
1
s
c st
N
N Q
M H
⎡ ⎤ ⎡ ⎤
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
= −⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥
⎢ ⎥ ⎢ ⎥−
⎣ ⎦ ⎣ ⎦
Q-H. Nguyen PhD Thesis Defense
40. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Governing Equations of Composite Steel-Concrete Beams
1 Equilibrium
Distributed bond
Discrete bond
2 Compatibility
3 Constitutive relations
H
scd
su
cu θ
θ
v
x
z
y
2
2
d ( )
( )
d
d ( )
( )
d
d
d ( )
( ) (
( )
) ( )
d
( )
d
c
c
s
sc s c
s
u x
x
x
u x
x
x
v x
x
v x
d x u x u x H
x
x
ε
ε
κ
=
=
=
+
−
= −
Matrix form
T
sc scd = ∂
= ∂e d
d
Q-H. Nguyen PhD Thesis Defense
41. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Governing Equations of Composite Steel-Concrete Beams
1 Equilibrium
Distributed bond
Discrete bond
2 Compatibility
3 Constitutive relations
[ ]nonlinear( ) ( )x f x=D e
( ) ( )x x=D k e
Section constitutive law
Section stiffness matrix
[ ]nonlinear( ) ( )sc scD x f d x=
( ) ( )sc sc scD x k d x=
Bond constitutive law
linear elastic
behaviour
Bond stiffness
Fiber discretization
of cross-section
y
z
linear elastic
behaviour
Q-H. Nguyen PhD Thesis Defense
42. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Exact Stiffness Matrix - Elastic behaviour
Distributed bond
Equilibrium in term of the displacements
5 3
2
15 3
3 4 2
2 33 4 2
2
4 2
d d
d d
d d d
d d d
d d
dd
s s
s s
s
c s
u u
x x
v u u
x x x
u v
u u H
xx
μ ζ
ζ ζ
ζ
⎧
− =⎪
⎪
⎪
= +⎨
⎪
⎪
= + +⎪
⎩
Analytical solution
compatibility relations
constitutive relations
Exact displacement fields
( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( )
)
) (
(s s
c c
s
c
v v
u x Z x
u x x Z x
x
v x x Z x
x x Z xθ θθ
= +
= +
= +
= +
X C
X C
X C
X C
Exact force fields
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( )
s s
c c
s N N
c N N
M M
T T
N x x Z x
N x x Z x
M x x Z x
T x x Z x
= +
= +
= +
= +
X C
X C
X C
X C ( ) ( ) 2
sinh cos( h 1 0 0) 0s x x xx xμ μ⎡ ⎤=
⎣ ⎦
X
Q-H. Nguyen PhD Thesis Defense
43. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Exact Stiffness Matrix - Elastic behaviour
Distributed bond
0= +eK q Q Q
1 1,Q q
2 2,Q q
3 3,Q q
4 4,Q q
5 5,Q q
6 6,Q q
7 7,Q q
8 8,Q q
L
1 8( 0) ... ( )
z
c
p
Q N x Q M x L= − = = =
↔ = +Q YC Q
Static boundary conditions
( )
1 8
1
( 0) ... ( )
z
c
p
q u x q x Lθ
−
= = = =
→ = −C X q q
Kinematic boundary conditions
Exact siffness matrix
Q-H. Nguyen PhD Thesis Defense
44. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Exact Stiffness Matrix - Elastic behaviour
Discrete bond
Composite beam element
with discrete bond
( )i
cu
( )i
su
( )i
θ
( )j
cu
( )j
su
( )j
θ
( )j
cu
( )i
v
( )i
cu
( )i
su
( )i
θ
( )j
cu
( )j
su
( )j
v
( )j
θ
( )j
su
( )j
θ
( )j
v
( )j
θ
+= +
( )i
v
( )i
cu
( )i
su
Connector
element
Unconnected
beam element
Connector
element
Q-H. Nguyen PhD Thesis Defense
45. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Exact Stiffness Matrix - Elastic behaviour
Discrete bond
Composite beam element
with discrete bond
( )i
cu
( )i
su
( )i
θ
( )j
cu
( )j
su
( )j
θ
( )j
cu
( )i
v
( )i
cu
( )i
su
( )i
θ
( )j
cu
( )j
su
( )j
v
( )j
θ
( )j
su
( )j
θ
( )j
v
( )j
θ
+= +
( )i
v
( )i
cu
( )i
su
Connector
element
Unconnected
beam element
Connector
element
nc
eK
Exact sitffness
matrix
Analytical
solution
Q-H. Nguyen PhD Thesis Defense
46. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Exact Stiffness Matrix - Elastic behaviour
Discrete bond
Composite beam element
with discrete bond
( )i
cu
( )i
su
( )i
θ
( )j
cu
( )j
su
( )j
θ
( )j
cu
( )i
v
( )i
cu
( )i
su
( )i
θ
( )j
cu
( )j
su
( )j
v
( )j
θ
( )j
su
( )j
θ
( )j
v
( )j
θ
+= +
( )i
v
( )i
cu
( )i
su
Connector
element
Unconnected
beam element
Connector
element
nc
eK
Exact sitffness
matrix
Analytical
solution
st
iK
Exact sitffness
matrix
Analytical
solution
st
jK
Exact sitffness
matrix
Analytical
solution
Q-H. Nguyen PhD Thesis Defense
47. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Exact Stiffness Matrix - Elastic behaviour
Discrete bond
Composite beam element
with discrete bond
( )i
cu
( )i
su
( )i
θ
( )j
cu
( )j
su
( )j
θ
( )j
cu
( )i
v
( )i
cu
( )i
su
( )i
θ
( )j
cu
( )j
su
( )j
v
( )j
θ
( )j
su
( )j
θ
( )j
v
( )j
θ
+= +
( )i
v
( )i
cu
( )i
su
Connector
element
Unconnected
beam element
Connector
element
nc
eK
Exact sitffness
matrix
Analytical
solution
st
iK
Exact sitffness
matrix
Analytical
solution
st
jK
Exact sitffness
matrix
Analytical
solution
Exact sitffness
matrix
eK
assembly
Q-H. Nguyen PhD Thesis Defense
48. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of the two bond models
20kN/m
6m 12m
40kN/m
12mmφ
800mm
100mm
IPE 200
200mm
80mm
Nelson 75-16
34GPa
210GPa
300000kN/m
1m
c
s
st
E
E
k
s
=
=
=
=
:stiffness of a single row of shears studs
:connector spacing
:equivalent distributed bond stiffness
st
sc
k
s
k
300MPast
sc
k
k
s
= =
Discrete bond model: using 18 elements
Distributed bond model: using 2 elements
Q-H. Nguyen PhD Thesis Defense
49. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of the two bond models
0 2 4 6 8 10 12 14 16 18
-20
0
20
40
60
80
100
120
140
160
180
200
Distance from left support [m]
Deflection[mm]
Discrete bond model
Distributed bond model
176 mm
180 mm
20kN/m40kN/m
Deflection distribution along the beam
Q-H. Nguyen PhD Thesis Defense
50. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of the two bond models
0 2 4 6 8 10 12 14 16 18
-1.5
-1
-0.5
0
0.5
1
1.5
2
Distance from left support [m]
Slip[mm]
Discrete bond model
Distributed bond model
20kN/m40kN/m
0.7−
1.1−
Slip distribution along the beam
Q-H. Nguyen PhD Thesis Defense
51. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of the two bond models
0 2 4 6 8 10 12 14 16 18
-0.03
-0.02
-0.01
0
0.01
0.02
Distance from left support [m]
Curvature[1/m]
Discrete bond model
Distributed bond model
20kN/m40kN/m
Curvature distribution along the beam
Q-H. Nguyen PhD Thesis Defense
52. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of the two bond models
0 2 4 6 8 10 12 14 16 18
-1000
-800
-600
-400
-200
0
200
400
600
800
1000
Distance from left support [m]
Axialforceintheconcreteslab[kN]
Discrete bond model
Distributed bond model
20kN/m40kN/m
Axial force distribution along the beam
Q-H. Nguyen PhD Thesis Defense
53. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of the two bond models
Conclusions
Discrete bond model: Discontinuities of axial force and curvature
Distributed bond model: all fields are continuous
Two distributed bond elements gives nearly identical results as
eighteen discrete bond elements
The discrete bond model represents the true connection and it is
simple to use but it requires a large number of elements
Q-H. Nguyen PhD Thesis Defense
55. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Time Effects in Concrete
1 Strain in concrete grow in
time
2 Shrinkage
3 Creep
4 Aging material
5 Play an important role in
serviceability
Curves of shrinkage, creep
and recovery after unloading
0t (start of drying)
loading
2t unloading1t
εsh = DRYING SHRINKAGE
ELASTIC RECOVERY
CREEP RECOVERY
σε ε ε= − sh
εsh
σ
εv = CREEP
εe= INITIAL ELASTIC STRAIN
t
t
t
εsh(t)
σε ( )t
ε( )t
Recovery
Load - free
Companion
Specimen
Loaded
(Creep)
Specimen
Specimen
Unloaded
σ
Q-H. Nguyen PhD Thesis Defense
56. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Linear viscoelastic model for concrete
Linear creep assumption: εc(t) = σcJ(t, t1)
Q-H. Nguyen PhD Thesis Defense
57. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Linear viscoelastic model for concrete
Linear creep assumption: εc(t) = σcJ(t, t1)
Principle of superposition in time (Boltzmann, 1874)
cε
1t
2t
2t
1σ
2σ
1 2σ σ+
2( )tε
1( )tε
cσ cε
cε
1t
cσ
cσ
1t
2t
2t1t
2( )tε
1( )tε
t
t
t
t
t
t
Q-H. Nguyen PhD Thesis Defense
58. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Linear viscoelastic model for concrete
Linear creep assumption: εc(t) = σcJ(t, t1)
Principle of superposition in time (Boltzmann, 1874)
cε
1t
2t
2t
1σ
2σ
1 2σ σ+
2( )tε
1( )tε
cσ cε
cε
1t
cσ
cσ
1t
2t
2t1t
2( )tε
1( )tε
t
t
t
t
t
t
Integral-type relation
εc(t) = σc(t1)J(t, t1) +
t
t1
J(t, τ)
dσc(τ)
dτ
dτ + εsh(t)
Q-H. Nguyen PhD Thesis Defense
59. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Time discrete approach
General method (step-by-step)
1
,
1
( ) ( )( ) ( ) ( )
n
c n sh n n i c i
i
c n c nE t t tt t εε σσ
−
=
≅ − + Ψ⎡ ⎤⎣ ⎦ ∑
2 1
1
,
1
1 1
1
( , ) ( , )
if i 1
( , ) ( , )
2
( )
( , ) ( , )
( , ) ( , )
if i 1
( , ) (
d
,
an
)
n n
n n n n
c n n i
n n n n
n i k i
n n n n
J t t J t t
J t t J t t
E t
J t t J t t
J t t J t t
J t t J t t
−
−
+ −
−
⎧ −
⎪ =
+⎪
⎪
= Ψ = ⎨
+ ⎪
−⎪ >
⎪ +⎩
[ ][ ]
1
1
1 1
1
d ( ) 1
( , ) d ( , ) ( , ) ( ) ( )
d 2
nt n
n n i n i i i
it
J t J t t J t t t t
σ τ
τ τ σ σ
τ
−
+ +
=
≅ + −∑∫
Trapezoidal rule
Time-discrete constitutive relation
where
Q-H. Nguyen PhD Thesis Defense
60. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Time discrete approach
Algebraic method (one step)
[ ]
1
1 1
d ( )
( , ) d ( ) ( )
d
( , ) with
nt
n n n
t
nJ t t tJt t tτ
σ τ
τ τ σ σ τ
τ
≅ − ≤ ≤∫
Effective modulus (EM) method (McMilan, 1916)
1
1
1
1 ( , )
(( , )) ,
( )
n
n
n
c
J
t t
J t t
E
t
t
ϕ
τ
+
= =
Time-discrete constitutive relation
1( ) ( ))( ) ( ( )c nc n c n sh n ct tE t t tσ ε ε σ≅ − + Ψ⎡ ⎤⎣ ⎦
[ ]1( , )
1
( , ) ( , )
2
n n nn J t t JJ t tt τ = +
Mean stress (MS) method (Hansen, 1964)
Age adjusted effective modulus (AAEM) method (Bažant, 1972)
1 1
1
1 ( , ) ( , )
( , )
( )
n
c
n
n
J
t t t t
E t
t
χ ϕ
τ
+
=
creep coefficient
aging coefficient
Q-H. Nguyen PhD Thesis Defense
61. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Time discrete approach
Rate-type (internal variables) method (Bažant, 1971)
0 1
1 1
( , ) 1 exp
( )
m
i ii
t
J t
E D
τ
τ
τ τ=
⎡ ⎤⎛ ⎞−
≅ + −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
∑
The creep function is approximated by the Dirichlet series
0 1 0
( ) 1
( ) ( ) with ( ) 1 exp
( )
tm
i i
i ii
t t
t t t d
E D
σ τ
ε ε ε τ
τ τ=
⎡ ⎤⎛ ⎞−
= + = −⎢ ⎥⎜ ⎟
⎝ ⎠⎣ ⎦
∑ ∫
The integral-type relation becomes
Time-discrete constitutive relation
( )shE εσ ε ε′Δ ′− Δ − ΔΔ=
0E
( ) ( ) ( )i i i iE t D t D tτ= −
( ) ( )i i it D tη τ= ( )m tη
( )mE t
1( )tη
1( )E t
( )tσ( )tσ
Aging Kelvin chain
Q-H. Nguyen PhD Thesis Defense
62. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Time-discretized analytical solution for composite beams
1 For one-step method
Faella et al. 2002
Ranzi and Bradford 2005
2 For rate-type (internal variables) method
Jurkiewiez et al. 2005
3 For general method
The solution is presented in the following
Q-H. Nguyen PhD Thesis Defense
63. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Time-discretized analytical solution for composite beams
1 For one-step method
Faella et al. 2002
Ranzi and Bradford 2005
2 For rate-type (internal variables) method
Jurkiewiez et al. 2005
3 For general method
The solution is presented in the following
Q-H. Nguyen PhD Thesis Defense
64. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Time-discretized analytical solution for composite beams
Time-discrete force-deformation relations
N(n)
s = (EA)s ε(n)
s
N(n)
c = (EA)(n)
c ε(n)
c − (EA)(n)
co ε
(n)
sh + (EB)(n)
c κ(n)
+
n−1
i=1
Ψn,i N(i)
co
M(n)
= (EB)(n)
c ε(n)
c − (EB)(n)
co ε
(n)
sh + (EI)(n)
κ(n)
+
n−1
i=1
Ψn,i M(i)
co
where
N(i)
co = α
(i)
1 x2
+ α
(i)
2 x + α
(i)
3 +
i
j=1
β
(i,j)
1 sinh(µjx) + β
(i,j)
2 cosh(µjx)
M(i)
co = α
(i)
4 x2
+ α
(i)
5 x + α
(i)
6 +
i
j=1
β
(i,j)
3 sinh(µjx) + β
(i,j)
4 cosh(µjx)
Q-H. Nguyen PhD Thesis Defense
65. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Time-discretized analytical solution for composite beams
Equilibrium in term of displacements
d5
u
(n)
s
dx5
− µ2
n
d3
u
(n)
s
dx3
= ζ
(n)
1 + ζ
(n)
2
n−1
i=1
Ψn,i
d2
M
(i)
co
dx2
+ ζ
(n)
3
n−1
i=1
Ψn,i
d2
N
(i)
co
dx2
d3
v(n)
dx3
= ζ
(n)
4
d4
u
(n)
s
dx4
+ ζ
(n)
5
d2
u
(n)
s
dx2
+ ζ
(n)
6
n−1
i=1
Ψn,i
dN
(i)
co
dx
Analytical solution
u(n)
s = X(n)
s C(n)
+ Z(n)
s (x) +
n−1
i=1
a(n,i)
s sinh(µix) + b(n,i)
s cosh(µix)
v(n)
= X(n)
v C(n)
+ Z(n)
v (x) +
n−1
i=1
a(n,i)
v sinh(µix) + b(n,i)
v cosh(µix)
Q-H. Nguyen PhD Thesis Defense
66. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Time-discretized analytical solution for composite beams
Time-discretized exact stiffness matrix
( )( )(
0
) ( )n nn n
= +e q Q QK
( ) ( )
1 1,n n
Q q
Exact siffness matrix at the instant
( ) ( )
2 2,n n
Q q
( ) ( )
3 3,n n
Q q
( ) ( )
4 4,n n
Q q
( ) ( )
8 8,n n
Q q
( ) ( )
5 5,n n
Q q
( ) ( )
6 6,n n
Q q
( ) ( )
7 7,n n
Q q
L
( ) ( )( ) ( )
1 8( 0) ... ( )n nn n
cq u x q x Lθ= = = =
Kinematic boundary conditions
( ) ( )( ) ( )
1 8( 0) ... ( )n nn n
cQ N x Q M x L= − = = =
Static boundary conditions
Composite beam element
nt
Q-H. Nguyen PhD Thesis Defense
67. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Applications
2300mm
200mm
934 mm
64.56kN/m
25m 25m
cG
sG
2
20 300mm×
2
1550 15mm×
2
30 450mm×
2
3
8 4
2
3
8 4
100mm
460000mm
0m
15.3310 mm
934 mm
42800mm
0m
159.4910 mm
c
c
c
c
s
s
c
s
H
A
S
I
H
A
S
I
=
=
=
=
=
=
=
=
666mm
Creep and shrinkage functions are defined in CEB-FIP Model Code 1990
0 0 28 030days, 30MPa, 80%, 196mm, 0.25sh ct t f RH h s= = = = = =
Two-span composite beam analyzed by Dezi and Tarantino, 1993
Q-H. Nguyen PhD Thesis Defense
68. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison with existing model
10
1
10
2
10
3
10
4
10
5
-2017
-2016
-2015
-2014
-2013
-2012
-2011
-2010
-2009
-2008
-2007
-2006
Time [days]
RedundantreactionR[kN]
Distributed bond - general method
Dezi and Tanrantino
64.56kN/m
25m 25m
= 0.4 kN/mm²sck
= 0.1kN/mm²sck
2007.16
2012.30
2013.51
2015.60
2007.72
2013.08
2016.30
2013.87
R
Time evolution of the redundant reaction at intermediate support
Q-H. Nguyen PhD Thesis Defense
69. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of the two bond models
0 10 20 30 40 50
0
5
10
15
20
25
30
Distance from left support [m]
Deflection[mm]
Discrete bond model (80 elements)
Distributed bond model (2 elements)
64.56kN/m
25550days
30days
Deflection distribution along the beam
Q-H. Nguyen PhD Thesis Defense
70. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of time-discrete approaches
10
1
10
2
10
3
10
4
10
5
17
18
19
20
21
22
23
24
Time [days]
Midspandeflection[mm]
Step-by-step method
"Rate-type" method
AAEM method
EM method
MS method
64.56kN/m
25m 25m
Creep effect only
Time evolution of midspan deflection
Q-H. Nguyen PhD Thesis Defense
71. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Effect of Creep and Shrinkage on Deflection
0 10 20 30 40 50
0
5
10
15
20
25
30
35
40
Distance from left support [m]
Deflection[mm]
30 days
25550 days: creep only
t=25550 days: creep + shrinkage
64.56kN/m
27%
39%
Deflection distribution along the beam
Q-H. Nguyen PhD Thesis Defense
72. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Effect of Creep and Shrinkage on Bending Moment
0 10 20 30 40 50
-10000
-8000
-6000
-4000
-2000
0
2000
4000
Distance from left support [m]
Bendingmoment[kN.m]
30 days
25550 days: creep only
t=25550 days: creep + shrinkage
70%
Bending moment distribution along the beam
Q-H. Nguyen PhD Thesis Defense
73. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Conclusions
An original time-discretized analytical solution has been derived
Compare to discrete bond model, the distributed bond model
leads to a quite complexe solution but it reduces significantly the
number of elements
General method gives precise results but it requires the storage of
the whole stress history
"Rate-type" method gives nearly identical results as general
method. This method avoids almost data storage but the
determination of model parameters is quite complexe
Among algebraic methods, AAEM method seems to perform very
well
A significant impact of shrinkage
Q-H. Nguyen PhD Thesis Defense
74. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Outline
4 Nonlinear Behaviour of Materials
Constitutive Models of Steel
Constitutive Models of Shear Stud
Constitutive Models of Concrete
Q-H. Nguyen PhD Thesis Defense
75. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Constitutive models of steel
1 Models based on explicit stress-strain relationship
2 Models based on plasticity framework
One-dimensional problem
Easy to implement for
monotonic loading
Cyclic models not easy to
formulate
Menegotto-Pinto model
( )1 0 r 1
ξ ε ε−
( )2 0 r 2
ξ ε ε−
0E 0E0E
hE
hE
( )0
,r rε σ
( )0 0 1
,ε σ
( )0 0 0
,ε σ
( )0 0 2
,ε σ
( )2
,r rε σ
( )1
,r rε σ
σ
ε
Q-H. Nguyen PhD Thesis Defense
76. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Constitutive models of steel
1 Models based on explicit stress-strain relationship
2 Models based on plasticity framework
( )e p
Eσ ε ε= −
Elastic stress-strain relationship
Yield condition and closure of the elastic range
Flow rule, isotropic and kineatic hardening laws
Kunh-Tucker complementarity conditions
Consistency condition
( ) ( )0 , , , 0 , , , 0f f XR RXλ σ λ σ≥ ≤ =
( ) ( ), , 0 if , , 0f fRX X Rλ σ σ= =
p
sign( )Xε λ σ= −
p λ=
sign( )Xα λ σ= −
Kinematic hardening stress-like variable
Kinematic hardening strain-like variable
( ) ( )(, 0), ( )yR R pf X X ασ σ σ= − − + ≤
Isotropic hardening strain-like variable
Isotropic hardening stress-like variable
Q-H. Nguyen PhD Thesis Defense
77. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Constitutive models of steel
1 Models based on explicit stress-strain relationship
2 Models based on plasticity framework
More involved
Cyclic behaviour is included in the model
Serveral physical phenomena can be coupled: damage, time effects ...
yε
σ
ε
yσ
uσ
hε uε
yε−hε−uε−
yσ−
uσ−
O 1O2O 3O4O
Monotonic loading
Cyclic loading
Linear isotropic hardening model
Q-H. Nguyen PhD Thesis Defense
78. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Constitutive models of shear stud
1 Discrete bond: The following models are selected to describe the
behaviour of one shear stud
P
δ
δ δ δ
P P P
uPuP uP
1 u0.95P P=
2 fu1.05P P=
fuP
uδ 1δ 2δ
( ) 2
u 11 exp
c
P P c δ= −⎡ ⎤⎣ ⎦
0E0E
0E
0E
Elastic-perfectly
plastic model
Ollgaard et al., 1971 Salari, 1999
2 Distributed bond: The equivalent distributed bond strength and
stiffness are calculated by dividing the strength and stiffness of a
single row of shear studs by their distance along the beam.
Q-H. Nguyen PhD Thesis Defense
79. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Constitutive models of concrete
Explicit stress-strain relationship recommended by CEB-FIP Model
Code 1990
Good approach of stress-strain
monotonic curve in compression
No unloading information
No ascending branch of stress-strain
monotonic curve in tension 1cE
c
E 1c
ε ,limc
ε
cm
f−
0.5 cm
f−
0.9 ctm
f
ctm
f
15%
cε
Compression
Tension
cσ
Goals: To develop a model for concrete based on the elasto-plastic
damage theory
1 Reproduce exactly stress-strain monotonic curve in compression
of CEB-FIP Model Code 1990
2 Take into account the degradation of the elastic moduli
3 Take into account the tension softening response
Q-H. Nguyen PhD Thesis Defense
80. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Constitutive models of concrete
Explicit stress-strain relationship recommended by CEB-FIP Model
Code 1990
Good approach of stress-strain
monotonic curve in compression
No unloading information
No ascending branch of stress-strain
monotonic curve in tension 1cE
c
E 1c
ε ,limc
ε
cm
f−
0.5 cm
f−
0.9 ctm
f
ctm
f
15%
cε
Compression
Tension
cσ
Goals: To develop a model for concrete based on the elasto-plastic
damage theory
1 Reproduce exactly stress-strain monotonic curve in compression
of CEB-FIP Model Code 1990
2 Take into account the degradation of the elastic moduli
3 Take into account the tension softening response
Q-H. Nguyen PhD Thesis Defense
81. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Governing equation of the elasto-plastic damage model
Assumption of a Helmholtz-free energy
Change of the compliance as internal variable (Govindjee at al., 1995)d
D
( ) ( )
1 2d 0 pdp p1
( , , ) ( )
2
DD Dp pεε ε ε
−
Ψ − = + − + Ψ
Elastic damage part plastic part
Thermodynamically associated variables
( )p
2
d
1ˆ; ;
2p
R Y
D
σ
ε ε
σ
∂Ψ ∂Ψ ∂Ψ
= = = = −
∂ − ∂ ∂
ε
σ
σ
p
ε
ε
0E
d
ε
e
ε
0EE
( )
1d d
D E
−
=
e
ε
( )
2d d d1
2
E εΨ =
( )
2e 0 e1
2
E εΨ =
Q-H. Nguyen PhD Thesis Defense
82. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Governing equation of the elasto-plastic damage model
Assumption of a Helmholtz-free energy
Change of the compliance as internal variable (Govindjee at al., 1995)d
D
( ) ( )
1 2d 0 pdp p1
( , , ) ( )
2
DD Dp pεε ε ε
−
Ψ − = + − + Ψ
Elastic damage part plastic part
Thermodynamically associated variables
( )p
2
d
1ˆ; ;
2p
R Y
D
σ
ε ε
σ
∂Ψ ∂Ψ ∂Ψ
= = = = −
∂ − ∂ ∂
( ) ( )ˆ, , ( ) 0yf R Y pRσ σ σ= − − ≤
Yield/damage condition
ε
σ
σ
p
ε
ε
0E
d
ε
e
ε
0EE
( )
1d d
D E
−
=
e
ε
( )
2d d d1
2
E εΨ =
( )
2e 0 e1
2
E εΨ =
Q-H. Nguyen PhD Thesis Defense
83. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Governing equation of the elasto-plastic damage model
Assumption of a Helmholtz-free energy
Change of the compliance as internal variable (Govindjee at al., 1995)d
D
( ) ( )
1 2d 0 pdp p1
( , , ) ( )
2
DD Dp pεε ε ε
−
Ψ − = + − + Ψ
Elastic damage part plastic part
Thermodynamically associated variables
( )p
2
d
1ˆ; ;
2p
R Y
D
σ
ε ε
σ
∂Ψ ∂Ψ ∂Ψ
= = = = −
∂ − ∂ ∂
Flow rule, damage and hardning/softening laws
( ) ( )ˆ, , ( ) 0yf R Y pRσ σ σ= − − ≤
Yield/damage condition
( ) ( )p d 1
1 1 sign( ) ; sign( ) ;
ˆ
f f f
D p
RY
ε λ λ σ λ λ σ λ λ
σ
β β β β
σ
∂ ∂ ∂
= − = − − = = − − = = −
∂ ∂∂
β : scalar paramater, proposed by Meschke et al, 1997
ε
σ
σ
p
ε
ε
0E
d
ε
e
ε
0EE
( )
1d d
D E
−
=
e
ε
( )
2d d d1
2
E εΨ =
( )
2e 0 e1
2
E εΨ =
Q-H. Nguyen PhD Thesis Defense
84. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Determination of the hardening/softening functions
Introducing a scalar parameter ζ
( )
0
1 p
E
ζ
σ
ε − = +
ε
σ
σ
0E 0EE
e
ε
p
ε d
ε
e
ε
d
E
( )p d
1 pζε ε+ = +
Assumption
Q-H. Nguyen PhD Thesis Defense
85. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Determination of the hardening/softening functions
Introducing a scalar parameter ζ
Monitonic loading condition
( ) ( )ˆ, , ( ) 0yf R Y R pσ σ σ= − + =
( )
0
1 p
E
ζ
σ
ε − = +
ε
σ
σ
0E 0EE
e
ε
p
ε d
ε
e
ε
d
E
( )p d
1 pζε ε+ = +
Assumption
Q-H. Nguyen PhD Thesis Defense
86. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Determination of the hardening/softening functions
Introducing a scalar parameter ζ
Monitonic loading condition
( ) ( )ˆ, , ( ) 0yf R Y R pσ σ σ= − + =
( )σ σ ε=
( )
0
1 p
E
ζ
σ
ε − = +
Explicit stress-strain relationship ε
σ
σ
0E 0EE
e
ε
p
ε d
ε
e
ε
d
E
( )p d
1 pζε ε+ = +
Assumption
Q-H. Nguyen PhD Thesis Defense
87. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Determination of the hardening/softening functions
Introducing a scalar parameter ζ
Monitonic loading condition
( ) ( )ˆ, , ( ) 0yf R Y R pσ σ σ= − + =
( )σ σ ε=
( )
0
1 p
E
ζ
σ
ε − = +
( )R R p=
Hardening/softening functions may be explicitly obtained
Explicit stress-strain relationship ε
σ
σ
0E 0EE
e
ε
p
ε d
ε
e
ε
d
E
( )p d
1 pζε ε+ = +
Assumption
Q-H. Nguyen PhD Thesis Defense
88. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Determination of the hardening/softening functions
In compression
In tension
( )
1
2 2 2
1 2 2 3 4
1 3
2 2 32 2
1 2
0 0 0
ˆ( )
1
ˆ cos arccos
3
c
i i i
i i i
i i i
R p p p p p p
p p p p p p
ζ ζ ζ ζ ζ
β β η η μ
+
−
= = =
⎡ ⎤
= − + + − − +⎢ ⎥
⎢ ⎥⎣ ⎦
⎧ ⎫⎡ ⎤⎛ ⎞
⎛ ⎞ ⎛ ⎞⎪ ⎪⎢ ⎥⎜ ⎟+ − + +⎨ ⎬⎜ ⎟ ⎜ ⎟⎢ ⎥⎜ ⎟
⎝ ⎠ ⎝ ⎠⎪ ⎪⎜ ⎟⎢ ⎥⎝ ⎠⎣ ⎦⎩ ⎭
∑ ∑ ∑
By using the stress-strain relationship of the
CEB-FIP Model Code 1990, we obtain
Hyperbolic softening law (Meschke et al., 1997)
2( )
1
ct
t
u
f
R p
p
p
=
⎛ ⎞
+⎜ ⎟
⎝ ⎠ 0E
ctf
σ
εctε
p
tR
relation σ ε−
( )tR p
0
d
01
E
E D+
tension
( )1
t
u
ct c
G
p
f l ζ
=
+
fracture energy
characteristic length
:tG
:cl
Q-H. Nguyen PhD Thesis Defense
89. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Integration algorithm for the elasto-plastic damage model I
1 Data at the time tn: {σn, εn, εp
n, pn, Dd
n}
2 Give at the time tn+1: ∆ε ⇒ εn+1 = εn + ∆ε
3 Predictor: compute elastic trial stress and test for inelastic
loading
σtrial
n+1 =
1
Dn
(εn+1 − εp
n)
Rtrial
= R(pn)
f trial
n+1 = σtrial
n+1 − Rtrial
IF f trial
n+1 ≤ 0 THEN
εp
n+1 = εp
n
pn+1 = pn
Dd
n+1 = Dd
n
σn+1 = σtrial
n+1
END → EXIT
ELSE proceed to step 4
Q-H. Nguyen PhD Thesis Defense
90. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Integration algorithm for the elasto-plastic damage model II
4 Corrector: by using the Kuhn-Tucker’s conditions, compute
∆λ and then update the other variables
∆λ = ∆p = pn+1 − pn
εp
n+1 = εp
n + (1 − β) ∆λsign(σtrial
n+1 )
σn+1 =
1
Dn
(εn+1 − εp
n) −
∆λ
Dn
sign σtrial
n+1
Dd
n+1 = Dd
n + ∆Dd
= Dd
n + β∆λ
sign(σtrial
n+1 )
σn+1
Compute the tangent modulus
Etg
n+1 =
∂σ
∂ε n+1
=
1
Dn+1
−
1
Dn+1 − (Dn+1)
2 ∂R
∂p n+1
END → EXIT
Q-H. Nguyen PhD Thesis Defense
91. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Numerical comparisons
-7-6-5-4-3-2-10
-30
-25
-20
-15
-10
-5
0
ζ
β
=
=
= −
=
0
Material parameters
27.9[MPa]
30000[MPa]
0.647
0.4
cmf
E
[mm/m]ε
[MPa]σ
Proposed model
Karsan and Jirsa, 1969
Simulation of cyclic compression test
0 0.1 0.2 0.3 0.4 0.5 0.6
0
1
2
3
4
[MPa]σ
[mm/m]ε
ζ
=
=
=
=
= −
0
Material parameters
3.5[MPa]
31000[MPa]
65[N/m]
68[mm]
0.2
ct
t
c
f
E
G
l
Proposed model
Gopalaratnam and Shah, 1987
Simulation of cyclic tension test
Good agreement of the calculated curve with the experiments is
observed
Q-H. Nguyen PhD Thesis Defense
93. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Displacement-Based Formulation
Assuming continuous displacement fields
Compatibility is satisfied in strict sense
Linearization of the constitutive equations
Equilibrium is satisfied in weak form
1q
2q
3q
4q
5q
6q
7q
8q
9q
10q
Element 10 DOF
( ) ( )x x=d a q
( ) ( ) ; ( ) ( )scx x d x x= = sce B q B q
1 1 1 1
;i i i i i i i
sc sc sc scD D k d− − − −
= + Δ = + ΔD D k e
( )T
d d 0 dsc sc e
L
D xδ δ∂ − ∂ − = ∀∫ D P
8 DOF: Xu and Aribert, 1995
10 DOF: Daniels and Crisinel, 1989
16 DOF: Dall'Asta and Zona, 2002
Q-H. Nguyen PhD Thesis Defense
94. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Displacement-Based Formulation
Assuming continuous displacement fields
Compatibility is satisfied in strict sense
Linearization of the constitutive equations
Equilibrium is satisfied in weak form
Governing equation of the finite element
1q
2q
3q
4q
5q
6q
7q
8q
9q
10q
Element 10 DOF
( ) ( )x x=d a q
( ) ( ) ; ( ) ( )scx x d x x= = sce B q B q
1 1 1 1
;i i i i i i i
sc sc sc scD D k d− − − −
= + Δ = + ΔD D k e
( )T
d d 0 dsc sc e
L
D xδ δ∂ − ∂ − = ∀∫ D P
0
1i
R
−
Δ = + −q Q Q QK
T 1 T 1
d di i
sc sc sc
L L
x k x− −
= +∫ ∫B k B BK BElement stiffness matrix
Element resisting forces T 1 T 11
d di i
s sc
L L
i
R cx D x−− −
= +∫ ∫B D BQ
8 DOF: Xu and Aribert, 1995
10 DOF: Daniels and Crisinel, 1989
16 DOF: Dall'Asta and Zona, 2002
Q-H. Nguyen PhD Thesis Defense
95. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Force-Based Formulation
Assuming continuous force fields
Equilibrium is satisfied in strict sense
0
( ) ( ) ( )
( ) ( ) ( ) ( )
sc sc sc sc
sc
D
x
x x x
x x x
= +
= + +
b Q c Q
D Db Q c Q
1Q
2Q
3Q
4Q
5Q
1scQ 2scQ 3scQ
/2L
( )scD x
( )scD x
/2L
Parabolic approximation of scD
zp
Discrete bond: exact force fields
Distributed bond: parabolic bond force distribution
(cubic approximation:
Salari 1999; Alemdar 2001)
2
2 0
d
0
d
d
0
d
d d
0
dd
c
sc
s
sc
sc
N
D
x
N
D
x
M D
H p
xx
+ =
− =
+ + =
Distributed bond
Element is internally determinate
Equilibrium Exact force fields
Element is internally indeterminate
A bond force distribution is assumed
Discrete bond ( )0scD =
Is a particular solution of equilibrium equations0( )xDEquilibrium equations
→
Q-H. Nguyen PhD Thesis Defense
96. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Force-Based Formulation
Assuming continuous force fields
Compatibility is enforced in a integral form
Linearization of the constitutive equations
Equilibrium is satisfied in strict sense
0
( ) ( ) ( )
( ) ( ) ( ) ( )
sc sc sc sc
sc
D
x
x x x
x x x
= +
= + +
b Q c Q
D Db Q c Q
1 1 1 1
;i i i i i i i
sc sc sc scd d f D− − − −
= + Δ = + Δe e f D
( ) ( )T T
d d 0 ,sc
L L
x D d x Dδ δ δ δ∂ − + ∂ − = ∀∫ ∫D d e d Dsc sc sc
1Q
2Q
3Q
4Q
5Q
1scQ 2scQ 3scQ
/2L
( )scD x
( )scD x
/2L
Parabolic approximation of scD
zp
Discrete bond: exact force fields
Distributed bond: parabolic bond force distribution
(cubic approximation:
Salari 1999; Alemdar 2001)
Q-H. Nguyen PhD Thesis Defense
97. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Force-Based Formulation
Assuming continuous force fields
Compatibility is enforced in a integral form
Linearization of the constitutive equations
Equilibrium is satisfied in strict sense
Governing equation of the force-based element
0
( ) ( ) ( )
( ) ( ) ( ) ( )
sc sc sc sc
sc
D
x
x x x
x x x
= +
= + +
b Q c Q
D Db Q c Q
1 1 1 1
;i i i i i i i
sc sc sc scd d f D− − − −
= + Δ = + Δe e f D
( ) ( )T T
d d 0 ,sc
L L
x D d x Dδ δ δ δ∂ − + ∂ − = ∀∫ ∫D d e d Dsc sc sc
1Q
2Q
3Q
4Q
5Q
1scQ 2scQ 3scQ
/2L
( )scD x
( )scD x
/2L
Parabolic approximation of scD
zp
Discrete bond: exact force fields
Distributed bond: parabolic bond force distribution
1
0
i
r
−
Δ = − − ΔQ qF q q
Element flexibility matrix
(cubic approximation:
Salari 1999; Alemdar 2001)
Q-H. Nguyen PhD Thesis Defense
99. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Two-field Mixed Formulation
Assuming continuous force fields
Slip compatibility is satisfied in strict sense
0( ) ( ) ( )x x x= + DD b Q
( ) ( )x x=d a q
Assuming continuous displacement fields
( ) ( )scd x x= scB q
Linearization of the constitutive equations
1 1 11
; i i i i
sc sc sc s
i i i
cD D k d−− − −
= + Δ= + Δe e f D Ayoub and Filippou 2000
2Q
zp
1Q
3Q 4Q
5Q
6Q
6 force DOF
1q
2q
3q4q
5q
6q
7q
8q9q
10q
10 displacement DOF
Q-H. Nguyen PhD Thesis Defense
100. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Two-field Mixed Formulation
Assuming continuous force fields
Slip compatibility is satisfied in strict sense
0( ) ( ) ( )x x x= + DD b Q
( ) ( )T T
d d d ,sc sc e
L L
D x xδ δ δ δ∂ − ∂ − + ∂ − ∀∫ ∫D P D d e d D
( ) ( )x x=d a q
Assuming continuous displacement fields
( ) ( )scd x x= scB q
Linearization of the constitutive equations
1 1 11
; i i i i
sc sc sc s
i i i
cD D k d−− − −
= + Δ= + Δe e f D
Equilibrium and section strain compatiblity are enforced in a integral
form (Hellinger Reissner variational principle)
Ayoub and Filippou 2000
2Q
zp
1Q
3Q 4Q
5Q
6Q
6 force DOF
1q
2q
3q4q
5q
6q
7q
8q9q
10q
10 displacement DOF
Q-H. Nguyen PhD Thesis Defense
101. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Two-field Mixed Formulation
Assuming continuous force fields
Slip compatibility is satisfied in strict sense
0( ) ( ) ( )x x x= + DD b Q
( ) ( )T T
d d d ,sc sc e
L L
D x xδ δ δ δ∂ − ∂ − + ∂ − ∀∫ ∫D P D d e d D
( ) ( )x x=d a q
Assuming continuous displacement fields
( ) ( )scd x x= scB q
Linearization of the constitutive equations
1 1 11
; i i i i
sc sc sc s
i i i
cD D k d−− − −
= + Δ= + Δe e f D
Equilibrium and section strain compatiblity are enforced in a integral
form (Hellinger Reissner variational principle)
Governing equation of the two-field mixed element
1 1
0
T 1
i i
sc e sc
i
r
− −
−
⎡ ⎤ ⎡ ⎤Δ + − −⎡ ⎤
⎢ ⎥ ⎢ ⎥=⎢ ⎥
⎢ ⎥ ⎢ ⎥− ⎢ ⎥⎣ ⎦⎣ ⎦ ⎣
Δ
⎦
K G q Q Q GQ Q
G F qQ 0
1
e
i
R
−
Δ = + −q Q Q QK
condense out
ΔQ
Ayoub and Filippou 2000
2Q
zp
1Q
3Q 4Q
5Q
6Q
6 force DOF
1q
2q
3q4q
5q
6q
7q
8q9q
10q
10 displacement DOF
Q-H. Nguyen PhD Thesis Defense
102. State determination algorithm: Displacement vs. Force models
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
Displacement-based element
Equilibrium?
Element resisting forces
T T
d di i
sc sc
L L
i
R x D x= +∫ ∫BQ B D
Constitutive laws
( ) ; ( )i i
R scR sc scD D d= =D D e
Compute deformations
( ) ; ( )i i
scx d xe
yes
Exit
no
1i i= +
General purpose finite element
program
Given displacements at the
structural nodes
Determinate resisting forces
and stiffness matrix
1i
g
−
q
1i
gR
−
P
1i
gU
−
ΔP
A
B
D
i
gΔq
gq
gP
1i
g
−
K
1
103. State determination algorithm: Displacement vs. Force models
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
Displacement-based element
Equilibrium?
Element resisting forces
T T
d di i
sc sc
L L
i
R x D x= +∫ ∫BQ B D
Constitutive laws
( ) ; ( )i i
R scR sc scD D d= =D D e
Compute deformations
( ) ; ( )i i
scx d xe
yes
Exit
no
1i i= + Element resisting forces
i
RQ
Constitutive laws
( ) ; ( )i i
R scR sc scD D d= =D D e
Compute deformations
( ) ; ( )i i
scx d xe
Compute element forces
;i i
scQ Q
Force-based element
Compute internal forces
( ) ; ( )i i
scx D xD
104. State determination algorithm: Displacement vs. Force models
Element resisting forces
i
RQ
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
Force-based element
Introduce an iteration scheme at the
element level
Consider element distributed loading
For regular beams: Spacone, 1994; Spacone
et al., 1996
For composite beams: Salari 1999; Alemdar 2001
Iteration sheme at the element level
No element internal loading
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
Force-based element
Itearative element
state determination
Nodal displacements
i
q
105. State determination algorithm: Displacement vs. Force models
Element resisting forces
i
RQ
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
Force-based element
Introduce an iteration scheme at the
element level
Consider element distributed loading
For regular beams: Spacone, 1994; Spacone
et al., 1996
For composite beams: Salari 1999; Alemdar 2001
Iteration sheme at the element level
No element internal loading
Propose a new state determination for composite
beam with element distributed loading
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
Force-based element
Itearative element
state determination
Nodal displacements
i
q
106. State determination algorithm for force-based element
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
Element displacements
1i i i−
= + Δq q q
n
gP
1n
g
+
P
101in
gg
−=+
Δ=ΔPP
1 0i n
g g
− =
=q q 1i
g
−
q i
gq 1n
g
+
q
1i
gR
−
P
1i
gU
−
ΔP
A
B
D
i
gΔq
Convergence
gq
gP
1i
g
−
K
1
Structure level
107. State determination algorithm for force-based element
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
1j i=
Δ = Δq q
Imposed displacements
Element forces
;j j
scQ Q
Element displacements
1i i i−
= + Δq q q
A
B
C
D
1i−
q i
q q
1j =
Δq
2j =
Δq
3j =
Δq
2j =
ΔQ
3j =
ΔQ
1i−
Q
i
Q
1j =
ΔQ
Q
1 1
1j =
K
1
1i−
K 2j =
K
j=3 convergence
1i−
q i
q q
1j =
Δq
2j =
Δq
3j =
Δq
scQ
1
12 2
sc sc
j j
sc
−= =
⎡ ⎤− Δ⎣ ⎦Q QF q
1
sc
i−
QK
1
sc
j =
QK
2
sc
j =
QK
1
1
13 3
sc sc
j j
sc
−= =
⎡ ⎤− Δ⎣ ⎦Q QF q
1j
sc
=
ΔQ
2j
sc
=
ΔQ
3j
sc
=
ΔQ
1i
sc
−
Q
i
scQ
j=3 convergence
A
B
C
D
Elementlevel
1
11 1
sc sc sc
j j j
j j j j j
sc sc
−
−− −
Δ = Δ
⎡ ⎤Δ = Δ − Δ⎣ ⎦Q Q Q
Q K q
Q K q F q
108. State determination algorithm for force-based element
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
1j i=
Δ = Δq q
Imposed displacements
Element forces
;j j
scQ Q
Internal forces
;j j
scDD
Element displacements
1i i i−
= + Δq q q
1
0
j j j
sc sc sc sc
j j i jj
sc
D
= =
+
+ + Δ
=
=
b Q c Q
D bQ cQ D
Particular solution due to the
element distributed loads
109. State determination algorithm for force-based element
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
1j i=
Δ = Δq q
Imposed displacements
Element forces
;j j
scQ Q
Internal forces
;j j
scDD
Deformations
;j j
scde
Constitutive laws
; ; ;j j j j
scR scRD fD f ;j j
scrr
Residual deformations
Element displacements
1i i i−
= + Δq q q
( )xD
1
( )xe
1j =
ΔD
1i−
D
i
D
2j =
ΔD
3j =
ΔD
1i−
e
1j =
Δe
i
e
2j =
Δe
3j =
Δe
1j
R
=
D 2j
R
=
D
1j =
r
2j =
r
1j =
f
2j =
f
1
1
1i−
f
A
B
C
D
1 1j j j j j j− −
Δ = Δ → = + Δe f D e e e
( )j j j j
R= −r f D D
Gauss-Labatto integration points
110. State determination algorithm for force-based element
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
1j i=
Δ = Δq q
Imposed displacements
Element forces
;j j
scQ Q
Internal forces
;j j
scDD
Deformations
;j j
scde
Constitutive laws
; ; ;j j j j
scR scRD fD f ;j j
scrr
Residual deformations 1j j −
⎡ ⎤= ⎣ ⎦K F
Element stiffness
Convergence?
,j j j j
scR scRD D tol− − ≤D D
Element displacements
1i i i−
= + Δq q q
111. State determination algorithm for force-based element
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
1j i=
Δ = Δq q
Imposed displacements
Element forces
;j j
scQ Q
Internal forces
;j j
scDD
Deformations
;j j
scde
Constitutive laws
; ; ;j j j j
scR scRD fD f ;j j
scrr
Residual deformations 1j j −
⎡ ⎤= ⎣ ⎦K F
Element stiffness
Element residual
1j+
Δq
displacements
Convergence?
,j j j j
scR scRD D tol− − ≤D D
Element displacements
1i i i−
= + Δq q q
no
next iteration j
( )
( )
T T
T T
1
1
d
dsc sc sc
j j
sc sc
L
j j j j
sc s
j
c
L
r x
r x
−
+
= − +
⎡ ⎤− ⎣ ⎦
Δ
+
∫
∫QQ Q Q
b r b
F F c r c
q
A
B
C
D
1i−
q i
q q
1j =
Δq
2j =
Δq
3j =
Δq
2j =
ΔQ
3j =
ΔQ
1i−
Q
i
Q
1j =
ΔQ
Q
1 1
1j =
K
1
1i−
K 2j =
K
j=3 convergence
Elementlevel
112. State determination algorithm for force-based element
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
1j i=
Δ = Δq q
Imposed displacements
Element forces
;j j
scQ Q
Internal forces
;j j
scDD
Deformations
;j j
scde
Constitutive laws
; ; ;j j j j
scR scRD fD f ;j j
scrr
Residual deformations 1j j −
⎡ ⎤= ⎣ ⎦K F
Element stiffness
Element residual
1j+
Δq
displacements
Convergence?
,j j j j
scR scRD D tol− − ≤D D
Element displacements
1i i i−
= + Δq q q
Element resisting forces
ji
R Q=Q
yes
no
next iteration j
113. State determination algorithm for force-based element
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
1j i=
Δ = Δq q
Imposed displacements
Element forces
;j j
scQ Q
Internal forces
;j j
scDD
Deformations
;j j
scde
Constitutive laws
; ; ;j j j j
scR scRD fD f ;j j
scrr
Residual deformations 1j j −
⎡ ⎤= ⎣ ⎦K F
Element stiffness
Element residual
1j+
Δq
displacements
Convergence?
,j j j j
scR scRD D tol− − ≤D D
Element displacements
1i i i−
= + Δq q q
Element resisting forces
ji
R Q=Q
Structure resisting forces
assemble( )i
R
i
R=P Q
1i
gU tol+
Δ ≤P
Convergence?
1i i
gU ext R
+
Δ = −P P P
Structure unbalanced forces
yes
no
next iteration j
114. State determination algorithm for force-based element
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
1j i=
Δ = Δq q
Imposed displacements
Element forces
;j j
scQ Q
Internal forces
;j j
scDD
Deformations
;j j
scde
Constitutive laws
; ; ;j j j j
scR scRD fD f ;j j
scrr
Residual deformations 1j j −
⎡ ⎤= ⎣ ⎦K F
Element stiffness
Element residual
1j+
Δq
displacements
Convergence?
,j j j j
scR scRD D tol− − ≤D D
Element displacements
1i i i−
= + Δq q q
Element resisting forces
ji
R Q=Q
Structure resisting forces
assemble( )i
R
i
R=P Q
1i
gU tol+
Δ ≤P
Convergence?
1i i
gU ext R
+
Δ = −P P P
Structure unbalanced forces
Exit yes
yes
no
next iteration j
115. State determination algorithm for force-based element
Structure equilibrium
-1
Solve i i i
g gUΔ = ΔK q P
1j i=
Δ = Δq q
Imposed displacements
Element forces
;j j
scQ Q
Internal forces
;j j
scDD
Deformations
;j j
scde
Constitutive laws
; ; ;j j j j
scR scRD fD f ;j j
scrr
Residual deformations 1j j −
⎡ ⎤= ⎣ ⎦K F
Element stiffness
Element residual
1j+
Δq
displacements
Convergence?
,j j j j
scR scRD D tol− − ≤D D
Element displacements
1i i i−
= + Δq q q
Element resisting forces
ji
R Q=Q
Structure resisting forces
assemble( )i
R
i
R=P Q
1i
gU tol+
Δ ≤P
Convergence?
1i i
gU ext R
+
Δ = −P P P
Structure unbalanced forces
Exit yes
yes
no
no
next iteration j
next NR iteration i
116. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison with experimental data
8φ
100
800
IPE 400
5 10φ
5 10φ
section A A
length unit: mm
650 650 650 650650 650 650 650
P
200 2500 2500 200
A
A
Poutre PI4
Load-deflection diagrams
Simply-supported composite beam (Aribert et al., 1983)
0 50 100 150
0
100
200
300
400
500
Midspan displacement [mm]
ForceP[kN]
18 Displacement-based elements
4 Force-based elements
4 Mixed element
Experiment (Ariber al al., 1983)
0 50 100 150
0
100
200
300
400
500
Midspan displacement [mm]
18 Displacement-based elements
12 Force-based elements
12 Mixed element
Experiment (Ariber al al., 1983)
33.3 mm30.7 mm
Distributed bond Discrete bond
Q-H. Nguyen PhD Thesis Defense
117. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison with experimental data
Slip distribution
0
0.5
1
1.5
Glissement[mm]
12 E.F. mixte
Résultat exprérimental
650 mm 650 mm 650 mm 650 mm650 mm 650 mm 650 mm 650 mm
P=257 kN
P=334 kNP=366 kN
P
-1.5
-1
-0.5
0
Glissement[mm]
6 E.F. mixte
Résultat exprérimental
P=297 kN
P=257 kN
P=297 kN
P=334 kN
P=366 kN
Discrete bond
9 connector element
Distributed bond
Q-H. Nguyen PhD Thesis Defense
118. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison with experimental data
P P
10 350× 10 350×
7 300×
100 2250 2250 2250 2250 100
B
B
C
C
Poutre PH3
10φ
100
800
HEA200
7.67 cm²
section CC
8.04 cm²
10φ
100
800
HEA200
1.6cm²
section BB
1.6cm²
Two-span composite beam (Ansourian 1981)
0 10 20 30 40 50 60
0
50
100
150
200
250
300
Midspan displacement [mm]
ForceP[kN]
24 Displacement-based elements
6 Force-based elements
6 Mixed element
Experiment
Load-deflection diagrams
Distributed bond
Q-H. Nguyen PhD Thesis Defense
119. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison with experimental data
-0.15 -0.1 -0.05 0 0.05 0.1 0.15
0
50
100
150
200
250
300
Courbures [1/m]
ForceP[kN]
24 E.F. déplacement
6 E.F. équilibre
6 E.F. mixte
Résultat exprérimental
P P
100 2250 2250 2250 2250 100
B
B
Poutre PH3
200
A
A
[ ]L : mm
Distributed bond
Section B-B
(negative bending)
Section A-A
(Positive bending)
Curvature [1/m]
ForceP[kN]
Q-H. Nguyen PhD Thesis Defense
120. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of the three finite element formulations
0p
2000mm
50mm50mm
20mm
yσ
sE
1
sE
1
σ
ε
yD
scE
1
scE
1
scD
scd
5
300MPa 2 10 MPa
200N/mm 1000MPa
y s
y sc
E
D E
σ = = ×
= =
A
A
section AA
Cantilever composite beam
Q-H. Nguyen PhD Thesis Defense
121. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of the three finite element formulations
2000mm
0p
δ
0 100 200
0
2
4
6
8
10
12
14
Deflection [mm]
1 element
2 elements
Converged solution
δ
0 100 200
0
2
4
6
8
10
12
14
Deflection [mm]
1 element
2 elements
Converged solution
δ
0 100 200
0
2
4
6
8
10
12
14
Deflection [mm]
Distributedload[kN/m]
1 element
2 elements
4 elements
64 elements
Converged solution
0p
δ
Displacement-based element Force-based element Mixed element
Load-deflection diagrams
Q-H. Nguyen PhD Thesis Defense
122. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of the three finite element formulations
0 500 1000 1500 2000
-20
-15
-10
-5
0
5
x [mm]
Bendingmoment[kN.m]
1 Displacement-based element
1 Force-based element
1 mixed element
Converged solution
0p 7 kN/m=
0 500 1000 1500 2000
-50
0
50
100
150
200
250
X [mm]
AxialforceNc[kN/m]
1 Displacement-based element
1 Force-based element
1 mixed element
Converged solution
0p 7 kN/m=
Poor representation of internal forces
Displacement-based & mixed models
Q-H. Nguyen PhD Thesis Defense
123. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Comparison of the three finite element formulations
0 500 1000 1500 2000
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
x [mm]
Curvature[1/m]
2 Displacement-based element
2 Force-based element
2 mixed element
Converged solution
0p 7 kN/m=
0 500 1000 1500 2000
0
0.05
0.1
0.15
0.2
0.25
x [mm]
Slip[mm]
2 Displacement-based element
2 Force-based element
2 mixed element
Converged solution
Force-based models
Inter-element slip discontinuity
Inter-element curvature discontinuity
Displacement-based & mixed models
Q-H. Nguyen PhD Thesis Defense
124. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Conclusions
Three finite element formulations have been developed for two
bond models
A new state determination algorithm for force-based element
included element distributed loads was presented
The numerical-experimental comparison shown validates the
models reliability and the capacity to determine the experimental
behaviour of composite beams
Force-based element and mixed element are both computationally
more efficient than the displacement-based element
For the same number of elements, force-based element yields
better results than mixed element
Q-H. Nguyen PhD Thesis Defense
125. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Outline
6 Time-Dependent Behaviour In the Plastic Range
Introduction
Viscoelastic/plastic Model for Concrete
Applications
Q-H. Nguyen PhD Thesis Defense
126. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Introduction
Viscoelastic models
Suitable to linear analysis only
Unable to account for the
cracking due to shrinkage
Viscoplastic models
Complicate to implement
10
1
10
2
10
3
10
4
-10
-8
-6
-4
-2
0
2
4
6
Time [days]
Stress[MPa]
Tensile strength: 2.9 MPa
200mm
100mm
( ) 0.03mmtδ ∀ = −
100mm
( )tσ
Concrete specimen C30, CEB-FIP model 1990
with shrinkage
without shrinkage
Stress relaxation according to viscoelastic model
Q-H. Nguyen PhD Thesis Defense
127. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Introduction
Viscoelastic models
Suitable to linear analysis only
Unable to account for the
cracking due to shrinkage
Viscoplastic models
Complicate to implement
→ propose a viscoelastic/plastic model
10
1
10
2
10
3
10
4
-10
-8
-6
-4
-2
0
2
4
6
Time [days]
Stress[MPa]
Tensile strength: 2.9 MPa
200mm
100mm
( ) 0.03mmtδ ∀ = −
100mm
( )tσ
Concrete specimen C30, CEB-FIP model 1990
with shrinkage
without shrinkage
Stress relaxation according to viscoelastic model
Q-H. Nguyen PhD Thesis Defense
128. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Viscoelastic/plastic Model for Concrete
Combination of linear visco-elasticity and
continuum plasticity (Van Zijl et al., 2001)
Decomposition of total strain
ve ve
( ) ( ) ( )t E t tσ ε σ= +
ve p sh
( ) ( ) ( ) ( )t t t tε ε ε ε= + +
viscoelastic
strain
plastic
strain
shrinkage
strain
Viscoelastic model Plastic model
Yield condition
( ) ( ), , ( ) 0yf R R pσ σ σ= − − ≤
p f
ε λ
σ
∂
=
∂
Flow rule
( )ve p sh
( ) ( ) ( ) ( ) ( )t E t t t tσ ε ε ε σ= − − +
0E
1( )E t 2 ( )E t
H
( )mE t
1( )tη 2 ( )tη ( )m tη
( )tσ ( )tσ
yσ
( )ve
tε ( )p
tε ( )sh
tε
Rheological viscoelastic/plastic model
Q-H. Nguyen PhD Thesis Defense
129. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Integration algorithm for Viscoelastic/plastic model I
1 Data at the time tn: {σn, εn, εp
n, pn, γ
(n)
i }
2 Give at the time tn+1: {εn+1, εsh
n+1}
3 Compute the parameters of the viscoelastic model: Eev
n+1, σn+1
4 Viscoelastic Predictor: compute viscoelastic trial stress and test
for plastic loading
σ trial
n+1 = Eve
n+1 εn+1 − εp
n − εsh
n+1 + σn+1
f trial
n+1 = f σtrial
n+1 , R(pn)
IF f trial
n+1 ≤ 0
THEN viscoelastic step :
εp
n+1 = εp
n
pn+1 = pn
σn+1 = σtrial
n+1
END → EXIT
ELSE plastic step: proceed to step 5
Q-H. Nguyen PhD Thesis Defense
130. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Integration algorithm for Viscoelastic/plastic model II
5 Corrector: by using the Kuhn-Tucker’s conditions, compute
∆λ and then update the other variables
pn+1 = pn + ∆λ
εn+1 = εn + ∆λsign(σtrial
n+1 )
σn+1 = σtrial
n+1 + Eve
n+1∆λsign(σtrial
n+1 )
Compute the tangent modulus
Etg
n+1 =
∂σ
∂ε n+1
= Eve
n+1
1 −
Eve
n+1
Eve
n+1 +
dR
dp
END → EXIT
Q-H. Nguyen PhD Thesis Defense
131. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Simulation of relaxation test
10
1
10
2
10
3
10
4
-10
-8
-6
-4
-2
0
2
4
6
Time [days]
Stress[MPa]
Linear viscoelastic model
Viscoelastic/plastic model
Tensile strength: 2.9 MPa
10
1
10
2
10
3
10
4
-0.08
-0.06
-0.04
-0.02
0
0.02
0.04
0.06
0.08
Time [days]
Strain[%]
Total strain
Viscoelastic strain
Plastic strain
Shrinkage strain
Time evolution of stress
Time evolution of strain
200mm
100mm
( ) 0.03mmtδ ∀ = −
100mm
( )tσ
Concrete specimen C30, CEB-FIP model 1990
The proposed model is able to
represent the cracking
phenomena due to shrinkage
Q-H. Nguyen PhD Thesis Defense
132. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Simply-supported composite beam
0 100 200 300 400 500
0
10
20
30
40
50
60
Force P [kN]
Flècheàmi-travée[mm]
A
=
= →
t 30 days
P 0 400kN
LoadP [kN]
Midspandeflection[mm]
10
1
10
2
10
3
10
4
0
10
20
30
40
50
60
Temps [jours]
Flècheàmi-travée[mm]
avec retrait
sans retrait
A
=
= →
P 400kN
t 30 days 50 years
Time [days]
Midspandeflection[mm]
without shrinkage effect
with shrinkage effect
8φ
100
800
IPE 400
5 10φ
5 10φ
section A A
length unit: mm
650 650 650 650650 650 650 650
P
200 2500 2500 200
A
A
Poutre PI4
Simply-supported composite beam (Aribert et al., 1983)
Evolution of mid-span deflection
Q-H. Nguyen PhD Thesis Defense
133. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Summary
1 A finite element model with exact stiffness matrix based on the
analytical solution was developed (for linear elastic and
viscoelastic behaviours)
2 A elasto-plastic damage model was proposed for concrete
3 Three finite element formulations was developed for composite
beams with partial interaction
4 A new state determination algorithm was developed for the
force-based element including element distributed load
5 A viscoelastic/plastic model was proposed for concrete in order to
simulate the interaction between the time effects and the cracking
Q-H. Nguyen PhD Thesis Defense
134. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Conclusions
1 The discrete bond model represents the true connection and it is
simple to use but it requires a large number of elements
2 Compare to discrete bond model, distributed bond model is less
computationally expensive because it reduces significantly
number of elements
3 Among three finite element formulations, force-based formulation
performs better
4 Significant influence of creep and especially of shrinkage on the
global response of composite beams in serviceability
5 Time effects play an important role in the inelastic response of
composite beam
Q-H. Nguyen PhD Thesis Defense
135. Introduction Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Conclusions
Future works
1 Realize a parametric study of time effects in nonlinear behaviour
of composite beam
2 Modelling of the behaviour of composite beams using
Timoshenko beam theory (in progress)
3 Take into account the nonlinearity geometry using corotational
formulation
4 Take into account the uplift
5 Extend the F.E. tools to composite frame
Q-H. Nguyen PhD Thesis Defense