Damage-based models for flexural response on RC elements

FINAL DOCTORAL EXAMINATION - May 2nd, 2012

DEPT. of CIVIL and ENVIRONMENTAL ENGINEERING at UC DAVIS
DEPT. of MECHANICS and MATERIALS at UNIVERSITY of REGGIO CALABRIA

DAMAGE-BASED MODEL FOR FLEXURAL RESPONSE OF
REINFORCED CONCRETE BEAM-COLUMN ELEMENTS

by
Maria Grazia Santoro

1. Objective and Scope of the Research
2. Overview of Previous Studies
3. Kinematic of Planar Frames
4. Typical Progression of Damage in RC Columns
5. Methodology, Hypotheses and Degradation Scheme
6. RC Damage-Based Beam Element
TABLE OF CONTENTS
TABLE OF CONTENTS

7. Damage Indices
8. Stiffness Matrix of the Damaged Member
9. Plastic functions: Yield Surface and Mixed Hardening Rule
• Integration Algorithm for Path-Dependent Plasticity
10. Energy Release Rate
11. Damage Criterion
12. Damage Evolution Law
• Integration Algorithm for Lumped Damage
13. Formulation of the Problem
14. Return Map Algorithm for Damage-Based Beam Element
15. Numerical Solution Strategies
• Structure State Determination
• Element State Determination
16. Numerical Examples
17. Remarks on Damage-Based Model Performance
18. Identification of Model Parameters dy and r

1. To implement a numerical model based on damage mechanics for predicting the
nonlinear flexural response of RC members under cyclic static loadings,
OBJECTIVE AND SCOPEOF THE STUDY
OBJECTIVE AND SCOPEOF THE STUDY

2. To predict damage indices by combining matrix structural analysis, plastic
theories and basic concepts continuum damage mechanics,
3. To introduce a different way of accounting for stiffness and strength degradation
based on damage indices,
4. To provide a new damage evolution law, calibrated on experimental tests by
using mechanical properties of the member,
5. The advantage of the model lies in its efficacy to evaluate the inelastic response
of RC members that couples computationally inexpensive finite elements with
classical moment-curvature analysis,
6. Damage indices introduced can facilitate the process of integrating structural
components damage to determine the overall structural performance,
7. The model and numerical simulations are implemented on a special purpose
computer program for nonlinear static and dynamic analysis of reinforced
concrete developed at University of California at Berkeley (FEDEASLab, Release
3.0) and in OpenSees environment,
8. Analytical results are compared with experimental results extracted from the
UW-PEER RC column performance database, which documents the performance
of more than 450 columns.

LUMPED PLASTICITY MODELS
the process of energy dissipation responsible for the nonlinear structural
response is concentrated at the ends of the finite element, in special
locations called plastic hinges
Clough [1965], Giberson [1967] and Otani [1974]
OVERVIEW OF PREVIOUS STUDIES
OVERVIEW OF PREVIOUS STUDIES

Filippou and Issa [1988], Kunnath et al. [1990] - increase of the inelastic zone length

LUMPED DAMAGE-BASED MODELS
interpreted as a combination between continuum damage mechanics and
the concept of plastic hinge
Cipollina et al. [1993] - Flòrez–Lòpez [1998]
Marante [2002] - Faleiro et al. [2008] - Alva et al. [2010]
Observation: use of force parameters to control damage accumulation and
tendency to overestimate damage and degradation at low to moderate
damage states
 
1  d  EA 0 0 
 a
L 


1  d i   4  d j  4 EI 4 1  d i  1  d j  2 EI 

K 0
 4  did j L 4  did j L 
 
 4 1  d i  1  d j  2 EI  1  d j   4  d i  4 EI 
 0
 4  did j L 4  did j L  
 ln 1  d  
d=  d a di dj  g  G   Gcr  q 
   1 d 

Local and global reference systems
and notation for planar frames
U=[ Uf Ud] P=[Pf Pd]
KINEMATIC OF PLANAR FRAMES

Uf unknown displ. – free DOFs

Ud assigned displ. – restrained DOFs
Pf nodal loads
Pd support reactions

Element nodal displacements Element nodal forces
u=[u1 u2 .... u6]T p=[p1 p2 .... p6]T
Basic element deformations Basic element forces
v=[v1 v2 v3]T q=[q1 q2 q3]T

COMPATIBILITY EQUATIONS
Element deformations can be expressed in terms of element end
displacements using the compatibility transformation matrix ag

  X Y X Y 

v  agu  L 
L
0
L L
0

 
U f  ag    Y X
1
Y

X
0
V  AU   A f Ad   
  U  L2 L 2
L2
L2 
 d   Y X Y X 
STRUCTURAL COMPATIBILITY MATRIX  2 0  1
 L 2
L L2
L2 

EQUILIBRIUM EQUATIONS
The equilibrium equations must be satisfied in the undeformed configuration, if
displacements are small relative to the structure.

p  bg q Nodal equations of static  Bf 
structural equilibrium P  BQ   Q B  AT
 Bd 
bg  aT applied forces P=[Pf Pd] are in equilibrium with
g
the resisting forces Pr, which are the sum of the P  Pr  0
element contributions p(el).

CONSTITUTIVE EQUATIONS – SECTION RESPONSE
KINEMATIC OF PLANAR FRAMES LINEAR ELASTIC RESPONSE OF FRAME ELEMENTS

 L 0 0
  EA 0 0

 EA   L 

L L 
 
4 EI

2 EI  q  f e 1  v  v 0 
fe   0  ke   0
 3EI 6 EI   L L  q  ke v  q0
 L L   2 EI 4 EI 
 0    0 
 6 EI 3EI   L L 
nonmechanical initial deformations v0, caused by temperature and shrinkage strains and fix–end
forces q0 under nonmechanical deformations

If displacements are large, equilibrium needs to be satisfied in the deformed
configuration. In this case nonlinear geometric effects must be taken into
account

accounting for P-∆ effects

using the matrix bP∆ instead
of bu
corotational formulation
(Crisfield, 1990)

p  bu q

NONLINEAR GEOMETRY and P-∆ GEOMETRIC STIFFNESS

a g  u  T q
k   k g   km  
  q  a g 

 u  a g  u 
u v  

element stiffness matrix in local coordinates is composed of two
contributions: the geometric stiffness kg arising from the change of the
equilibrium matrix with end displacements, and the material stiffness
km, which represents the transformation of the tangent basic stiffness
matrix to the local coordinate system

1. Increasing of axial deformations increase strains of the concrete cover until
PROGRESSION OF DAMAGE IN RC COLUMNS
PROGRESSION OF DAMAGE IN RC COLUMNS cracking and spalling. The loss of cross-sectional area imposes additional stresses
on the remaining concrete core and on longitudinal reinforcement (Bresler, 1961).
2. The longitudinal reinforcement yields in tension and eventually begins to strain
harden.
3. Poisson’s effect causes expansion of the concrete core, which exerts pressure on
both longitudinal and transverse reinforcement.
4. Transverse reinforcement restrains the lateral deflection of longitudinal
reinforcement, and it confines the expanding core. The confining pressure is not
uniform: it depends on stiffness and strength of transverse reinforcement (Bresler
and Gilbert, 1961). Additionally, the stiffness of the tie depends on its strain, which
in turn is affected by the axial deformation of the column and by bar buckling.
5. The increased axial strain and imposed lateral deformations (due to core
expansion) lead to instability of longitudinal bars (Bayrak and Sheikh, 2001).
When the tie spacing is very large, longitudinal bar buckling can occur between
two adjacent ties (Dhakal and Maekawa, 2002). In other situations, bar buckling
can occur over several tie spacings.
6. Bar buckling is affected by the maximum tension strain and by tension strain
growth (associated with cyclic inelastic deformations) in the longitudinal
reinforcement (Moyer and Kowalsky, 2001).
7. The effect of cycling on the constitutive properties of the concrete and steel, is
significant (Monti and Nuti, 1992). The load history and cycling affect damage
progression and specifically bar buckling (Kunnath et al., 1997).

FUNDAMENTALS OF CONTINUUM DAMAGE MECHANICS
DEFINITION OF DAMAGE
degradation of material properties resulting from initiation,
growth and coalescence of microcracks or microvoids, and it is
METHODOLOGY – HYPOTHESES

represented by a scalar in the range 0-1
DEGRADATION SCHEME
DEGRADATION SCHEME

Sd
d
S
EFFECTIVE STRESS
the ratio between the load applied on the volume element and
the effective resistance area


1  d 
HYPOTHESIS OF STRAIN EQUIVALENCE
the strain associated with a damage state under the Cauchy
stress is equivalent to the strain associated with its undamaged
state under effective stress
 
 
e

E E 1  d 

The modelling of reinforced concrete behaviour comes from the
experimental observation that damage is a continuous process that initiates
at very low levels of applied loads and leads to an increasing amount of
damage when levels of strain increase. Conversely, the behavior of steel
bars is dominated by plasticity laws and damage materializes at higher

deformation levels
DEGRADATION SCHEME
DEGRADATION SCHEME

For reinforced concrete structures, plasticity is physically associated to the flow
of reinforcement, while damage indicates cracking and crushing of the concrete

CONCENTRATED PLASTICITY-DAMAGE APPROACH
simplicity, computational convenience of having stiffness matrix in a
concise form
FLEXURAL RESPONSE UNDER UNIAXIAL BENDING and
CONSTANT AXIAL LOAD

hysteretic behavior of RC is governed by only few parameters
DEGRADATION SCHEME
DEGRADATION SCHEME

SMALL DEFORMATIONS
SHEAR DEFORMATIONS NEGLECTED
AXIAL BEHAVIOR LINEAR ELASTIC AND UNCOUPLED FROM THE
FLEXURAL BEHAVIOR
simple truss element w/o second order effects
UNCOUPLED PLASTIC-DAMAGE BEHAVIOR
Independent constitutive eqns.  (1) change of the elastic properties of
the material is produced only by damage; (2) plasticity only produces
incompatible strains

combined degradation of both strength and stiffness under the
effect of damage indices and a mixed-hardening rule
DEGRADATION SCHEME
DEGRADATION SCHEME

The softening effect resulting into strength loss is caused by damage,
meaning that an increment of damage at a constant plastic deformation,
produces a decrease of the plastic limit. Damage also controls the
degradation of the unloading stiffness.

ONE COMPONENT MODEL
RC DAMAGE-BASED BEAM ELEMENT
RC DAMAGE-BASED BEAM ELEMENT

element forces = B. MOMENTS
q=[qi qj]T
element deformations= ROTATIONS
v=[vi vj]T

q  qel  q pl
v  v el  v in
q  kel v el  kel  v  v in 
v in  v pl  v d

d=[di dj]
Numerical quantification of flexural damage at hinges i and j
Damage parameters take values in the interval [0 1]
If d=0 (no damage),  standard plastic hinge
If d=1 (hinge totally damaged)  internal hinge in elastic member

d+ =  d+
 i d 
j  d  = d 
 i d j 

DAMAGE INDICES
DAMAGE INDICES

for RC members subjected to cyclic loadings two different set of cracks can
appear, one due to positive end moments (positive cracks) and another due
to negative end moments (negative cracks).
This behavior can be represented by using two sets of damage variables
where the superscript + and – denote damage due to positive and negative
moments

 2   4 
2
3 1- d i  1- d j  EI  1- d j  
STIFFNESS MATRIX OF DAMAGED MEMBER
STIFFNESS MATRIX OF DAMAGED MEMBER
 -1 
L  1- d i    
f ed  ked 
6 EI  -1

2 

4  1- d i  1- d j  L  4 
 1- d j    2 

  
 1- d i  


for di=dj=0, the stiffness
matrix of the damaged
member is equal to the
elastic stiffness matrix.
If di=1 and dj=0, then ked
becomes the stiffness
matrix of an elastic
member with an internal
hinge at the left end.
If di=0 and dj=1, then ked
becomes the stiffness
matrix of an elastic
member with an internal
hinge at the right end.

The yield function is obtained according to a mixed-hardening rule

YIELD SURFACE and MIXED HARDENING RULE
which control the rate of transition of the yield surface.
YIELD SURFACE and MIXED HARDENING RULE If isotropic and kinematic hardening are combined, the yield surface is
allowed both to expand and to translate, providing a more realistic
modeling of the real behavior, especially when dealing with cyclic loading.

f  q , qb ,    n q  qb   q pl  H is   0
T
PLASTIC FUNCTIONS:
PLASTIC FUNCTIONS:

q=  q i q j 
 
q b = q bi q bj 
 
q pl =  M yi M yj 
 = i  j 

f sign  qi  qbi  0 
n  ni n j  
T
 
  q

 0 sign  q j  qbj  


FLOW RULE and HARDENING RULES

v pl   sign  q  qb 
INTEGRATION ALGORITHM FOR PATH-


   , qb  H k v pl
  
DEPENDENT PLASTICITY

KUHN-TUCKER CONDITIONS

k  0, f k  0, k f k  0, k fk  0, k  i, j

 
 0 if f  0 (no plasticity)
 0 if f  0 (plastic increment)

CONSISTENCY CONDITIONS

f < 0 if   0
 f = 0
  0 if f = 0

DISCETIZATION of GOVERNING EQNS.

f  q , qb ,  
  T
n  kel  H k  n  H is

kel nn T kel
kep  ke  T
n  kel  H k  n  H is

g  G , R , d   Gt  , d   Rt  d   0

G = ENERGY RELEASE RATE
 It is a function of the independent variable t, the so-called pseudo-time
DAMAGE CRITERION
DAMAGE CRITERION

(for cyclic static loadings is the current loading step);
 It is a function of plastic curvatures;
 It is a function of d, the damage internal variable.

R = DAMAGE THRESHOLD
 It is a function of d, the damage internal variable;
 R0 ≡ Gcr is the initial damage threshold, that is the amount of energy
stored when the first crack forms  Rt>R0
 It is governed by a time law of evolution

damage is initiated when the damage energy release rate G
exceeds the initial damage threshold Gcr

G = energy stored during loading/unloading process
   e , p , d   1  d   0   e , p 

p is a set of internal plastic
parameters, corresponding to
ENERGY RELEASE RATE
ENERGY RELEASE RATE

isotropic and kinematic plastic
hardening variables (namely, β and α)

For uncoupled behavior
 0   e , p    el   e , d    pl  p 

1 T
   , d    e ked  d   e   pl   pl 
0 = i j 
 
2


G :   0   e , p 
d

ENERGY RELEASE RATE conjugated to the damage variable d

DISCETIZATION of ENERGY RELEASE RATE G

 Mk 
  1  Mk 
 
  k , pl 

Gk     L pl   
 1  dk   2 EI  1  d k  
Gi Gi 
G d     
G j G 
ENERGY RELEASE RATE

 j  L pl  0.08 L  0.022 f yd d b
ENERGY RELEASE RATE

 pl     y  L pl

at the current step the total
energy release rate is equal to
the energy stored at the
previous step Gn, incremented
by the plastic contribution at
the current step ∆Gn+1,
proportional to bending
moment and deformation
increment (curvature), and
extended to the inelastic zone
Lpl

G at the n-th step is the
summation of the initial
elastic stored energy Gel and
Gn+1=Gn+∆Gn+1 the plastic contribution

SOFTENING RULE
The identification of an appropriate softening rule
is the most important requirement for the complete
   g k
d k k k  i, j definition of the damage-based model. It is related
to the mechanical properties of the RC member,
Gk and accounts for cracking, yielding and ultimate
bending, as well as inelastic hinge deformations.
DAMAGE EVOLUTION LAW

KUHN-TUCKER CONDITIONS d i d i 
D  
d
k  0, g k  0, k g k  0 d j
 j 

CONSISTENCY CONDITIONS

 0 if g k  0  no damage 

k g k  0
 k 

 0 if g k  0  damage increment 


The softening rule is given in a numerical form to enable representation of
the physical evolution of damage in the member. Test results as well as
numerical simulations (Shah; Chung et al.; Park et al; Gupta et al.; Sadeghi
et al.) show that cumulative damage increases as a function of
deformations rather than forces

DISCETIZATION of DAMAGE EVOLUTION LAW R

Ri Ri   M y , k 1  bk  
R d     
Rk ( d k )  Gcr , k   ak  1  k
d
R j R j 
   1   d k r  r 
   

M u ,k  M y ,k
Sin ,k   tg 2
du  d y

M y ,k
S f ,k   tg1
dy

Sin ,k ak  M y ,k  bk
b
S f ,k
2
M cr
R0  Gcr  L pl
2 EI

PREDICTED RATE OF DAMAGE AT YIELDING – dy

d y = 0 .0 1 d y = 0 .1 d y = 0 .2 d y = 0 .3 d y = 0 .5
R

0 0 .1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7 0 .8 0 .9 1
d

dY SHIFTS THE INTERSECTION POINT D
OF THE TWO TANGENTS tg1 AND tg2 du = 1

EXPONENT OF THE FUNCTION – r
5
r=1.5 r=2 r=4 r=7 r=10
4

4

3

3
R

2

2

1

1

5

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 d 1

r CONTROL THE SHAPE OF THE FUNCTION R(d)
r>0

INTEGRATION ALGORITHM FOR LUMPED DAMAGE
INTEGRATION ALGORITHM FOR LUMPED DAMAGE
g i  Gi  Ri  0
 g i g i   max  1,:    g i ,trial 
g i  Gi  Ri  0   
g trial    g 
g j gj  max   2,:    j ,trial 
 
g   G  R   0
j j j
ONLY ONE DAMAGE FUNCTION G IS NEEDED
g   G  R   0
j j j FOR EACH HINGE TO CHARACTERIZE THE STATE
OF DAMAGE, WHETHER THIS IS DUE TO
POSITIVE OR NEGATIVE BENDING MOMENT

DETERMINATION of the DAMAGE INCREMENT

d  G  d  - R  d  dg trial
  d n 1  d n  d n 1
subscripts i and j and superscript + and – are omitted for simplicity

Rk S f , k 1  bk 
Stg , k   S f , k bk  1 r
d k
1   d k r 
 
r

Given:
(i) The geometry of the structure defined by nodes coordinates, and
FORMULATION OF THE PROBLEM

the connection table that identifies each member,
FORMULATION OF THE PROBLEM

(i) The mechanical properties of each member,
(ii) The relation moment-curvature at each hinge
(iii) The loading history of nodes during pseudo-time interval [Tmin Tmax],
(iv) The displacement history of nodes during a pseudo-time interval [Tmin Tmax],
Calculate:
(i) Displacement history of nodes during at each loading step,
(ii) Reactions on nodes (base shear),
Calculate and update at the end of every load step (n+1):
(i) The damaged stiffness matrix of the structure
(ii) Basic element force vector q, basic element deformations v, plastic deformation
vpl, back stresses qb, damage indices collected in the matrix D and all the
remaining internal variables α and β, for each member of the structure
Such that they verify:
(i) Compatibility equations,
(ii) Equilibrium equations,
(iii) Yield functions f=[fi fj]<0 and damage functions at nodes i and j, gtrial=[gi,trial
gj,trial]>0
(iv) Internal variables evolution laws.

RETURN MAP ALGORITHM
RETURN MAP ALGORITHM

The state determination process is made up of two nested phases:

a) THE ELEMENT STATE DETERMINATION: the element resisting
NUMERICAL SOLUTION STRATEGIES

forces are determined for the given element end deformations

b) THE STRUCTURE STATE DETERMINATION: the element resisting
forces are assembled to the structure resisting force vector

The resisting forces are then compared with the total applied loads and
the difference, if any, yields the unbalanced forces which are then
applied to the structure in an iterative solution process until external
loads and internal resisting forces agree within a specified tolerance.

INCREMENTATION STEP k – Advancing phase
The external reference load pattern Pref is imposed as a sequence of load
increments ΔPfk = ∆λPref, where Δλ is is the load factor increment. At load step k
the applied load is equal to Pfk =Pfk-1+∆Pfk, with k=1,2…nstep and Pf0=0

ITERATION STEP t – Correcting phase
This iteration loop yields the structural displacements Uk that correspond to
applied loads Pfk

THE ELEMENT STATE DETERMINATION - s

k s=0 =k t 1

Final point

Unbalanced deformation
Starting point
v s=2 =v t -v s=2
u r

residual deformation

v s=2 =v s=2 +v s=2
r el pl

Given deformation


STATE
STRUCTURE

DETERMINATION

NUMERICAL EXAMPLES
NUMERICAL EXAMPLES

NUMERICAL EXAMPLES
NUMERICAL EXAMPLES

Reinforcement ratios are in the range 0.6-3.6%, while axial load ratios vary from 0 to 0.56
An average value of EI applicable to the entire length of the member was estimated by correcting the
stiffness of the uncracked member through parameter γ

Normal strength (NSC) and high strength (HSC) specimens

For normal-strength concrete, the model proposed by Hoshikuma et al. was used for determining the
confined properties, while the Muguruma et al. model was employed for high-strength concrete

Moment-curvature analyses were performed using OpenSees using a zero-length element. Sections
were discretized into confined and unconfined concrete regions, for which separate fiber
discretizations were generated. Reinforcing steel bars were placed around the boundary of the
confined and unconfined regions. Sections were discretized into ten layers inside the concrete core and
two layers for the unconfined concrete on each side.

For reinforcing steel, a bilinear stress-strain relationship was used, with elastic modulus Es=210 GPa
and 1% strain-hardening ratio.

Numerical simulations were performed under force control, imposing forces at the free end of the
member in an iterative manner as to induce the desired displacement history

200

150

Moment-Curvature Analysis of the cross-
M [KNm]

100 section – The section shows hardening behavior;
the estimated yielding and ultimate curvatures are
50 ϕy=0.0066 and ϕu=0.07351, respectively

0
0 0.02 0.04 0.06 0.08 0.1 0.12
 [1/m]

150

100

50
Shear base [KN]

0

-50

-100
model test
-150
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
Top Displacement [m]
Force-Displacement History – The results of the numerical simulation are depicted in blue. From the comparison with the
experimental results represented with a dashed green line, it can be observed that the damage-based model is able to provide a
good agreement with the test regarding the evaluation of the strength peaks during the loading/unloading process and an
adequate estimation of the initial unloading stiffness

150

100

50
Shear force [KN]

0

-50

-100
model test
-150
0 100 200 300 400 500 600 700
iterations

Shear Force Development during the Loading Cycles – It can be observed that the first cycle shows some
discrepancies between analysis and experiment, because the effect of the first cycle was not included in the analysis and the
difficulty of establishing the right initial conditions for the model. The likelihood of an abnormal behavior increases with the
number of load cycles (iterations) because of the monotonically increasing value of the energy dissipation, which affects the
degradation parameter d

200

150

100
bending moment M,Qb

50

0

-50

-100

-150 M- qb-pl test
-200
-0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
rotations ,
pl

Bending Moment-Rotation History – The green line represents the progression of the back stress with the plastic
deformation θpl. The slope of the back stress is the kinematic hardening ratio of the member which controls the
increasing of plastic deformation during the cyclic loading process. The degradation of the elastic properties instead is
produced only by the damage index d since the damage-based model is uncoupled

g
140
Threshold Energy g=G-R
120

100

80
R(d),G(d),g(d)

60

40

20

0

-20
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Positive Damage index
Progression of the Damage Functions with the damage index d – The damage threshold R shows an increasing
exponential behavior governed by the damage index d and the exponent of the damage evolution law r=1.15. The energy
function G represents the amount of energy stored during the loading process and it is a typical step function. The
constant energy intervals correspond to the unloading branches since the unloading process in the damage-based model
is elastic. The initial damage threshold is Gcr=0.0022 [KNm] and it corresponds to the formation of the first crack in the
member. The function g represents the damage criterion adopted for the model, g=G–R>0

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0
0 100 200 300 400 500 600 700
No. of iterations
Damage Index Evolution – The progression of the damage index with the load cycles is captured in this graph. The damage
index is recorded only for positive bending at the bottom of the cantilever column, where the plastic hinge forms. The damage
index starts to grow after the first crack forms in the member and progresses very slowly until the first yielding of the member
itself. After that, the progression of the damage becomes steeper until failure. The maximum value of damage is dmax=0.4251

Ductility factors – The graph reports the ductility of the member for every load cycle. The cumulative ductility of the
member is also estimated to μcum=41.05. The displacement at first yielding is approximated to δy=0.01032 [m]

900

800

700 Moment-Curvature Analysis of the cross-section –
600 The section shows softening behavior thus the failure is
governed by the crushing of the concrete core; the
M [KNm]

500

400 estimated yielding and ultimate curvatures are ϕy=0.013
300 and ϕu=0.1977, respectively
200

100

0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
 [1/m]

800

600

400
Shear base [K ]

200
N

0

-200

-400

-600
model test
-800
-0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
Top Displacement [m]


Shear Force Development during
800
the Loading Cycles – It can be
600
observed that the first cycle shows
some discrepancies between analysis
400 and experiment, because of the
difficulty of establishing the right initial
Shear force [KN]

200 conditions for the model. The
likelihood of an abnormal behavior
0
increases with the number of load
-200
cycles (iterations) because of the
monotonically increasing value of the
-400 energy dissipation, which affects the
degradation parameter d. Furthermore,
-600 due to the application of the load
model test
control method the nonlinear behavior
-800
0 100 200 300 400 500 600 700 near the ultimate strength peaks of the
iterations member is not well matched

1500
Bending Moment-Rotation History
1000 – The green line represents the
progression of the back stress with the
bending moment M,Qb

500 plastic deformation θpl. The slope of
the back stress is the kinematic
hardening ratio of the member which
0
controls the increasing of plastic
deformation during the cyclic loading
-500 process. The degradation of the elastic
properties instead is produced only by
-1000 the damage index d since the damage-
M- qb-pl test based model is uncoupled
-1500
-0.06 -0.04 -0.02 0 0.02 0.04 0.06
rotations ,pl

Progression of the Damage Functions
700 with the damage index d – The damage
Threshold Energy g=G-R threshold R shows an increasing exponential
600 behavior governed by the damage index d
500
and the exponent of the damage evolution
law r=1.30. The energy function G
400 represents the amount of energy stored
R(d),G(d),g(d)

during the loading process and it is a typical
300
step function. However, in this particular
200 case, due to the relatively high number of
iterations, it is not possible to distinguish the
100 constant energy intervals. The initial damage
threshold is Gcr=0.0576 [KNm] and it
0
corresponds to the formation of the first
-100 crack in the member. The function g
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Positive Damage index represents the damage criterion adopted for
the model, g=G–R>0

0.5
Damage Index Evolution – The
progression of the damage index with the
0.4 load cycles is captured in this graph. The
damage index is recorded only for positive

0.3 bending at the bottom of the cantilever
column, where the plastic hinge forms.
The damage index starts to grow after the
0.2 first crack forms in the member and
progresses very slowly until the first
yielding of the member itself. After that,
0.1
the progression of the damage becomes
steeper until failure. The maximum value
0 of damage is dmax=0.4816
0 100 200 300 400 500 600 700
No. of iterations

Ductility factors – The graph reports the ductility of the member for every load cycle. The cumulative ductility of the
member is also estimated to μcum=32.75. The displacement at first yielding is approximated to δy=0.01408 [m]

50

45

40
Moment-Curvature Analysis of the cross-section –
35
The section shows softening behavior thus the failure is
30
governed by the crushing of the concrete core; the
M [KNm]

25
estimated yielding and ultimate curvatures are ϕy=0.023
20
and ϕu=0.2809, respectively
15

10

5

0
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
 [1/m]
80

60

40
Shear base [KN]

20

0

-2 0
m od el
-4 0
te s t
-6 0

-8 0
-0 .0 6 -0 .0 4 -0 .0 2 0 0 .0 2 0 .0 4 0 .0 6
T o p D is p la c e m e n t [m ]

Shear Force Development during
80
the Loading Cycles – It can be
60 observed that the first cycle shows
some discrepancies between analysis
40 and experiment, because of the
difficulty of establishing the right initial
Shear force [KN]

20 conditions for the model. The
likelihood of an abnormal behavior
0
increases with the number of load
-20 cycles (iterations) because of the
monotonically increasing value of the
-40 energy dissipation, which affects the
degradation parameter d. Furthermore,
-60 due to the application of the load
model test
control method the nonlinear behavior
-80
0 200 400 600 800 1000 1200 1400 1600 1800 2000 near the ultimate strength peaks of the
iterations member is not well matched

50

Bending Moment-Rotation History
– The green line represents the
progression of the back stress with the
b
bending moment M,Q

plastic deformation θpl. The slope of
the back stress is the kinematic
0 hardening ratio of the member which
controls the increasing of plastic
deformation during the cyclic loading
process. The degradation of the elastic
properties instead is produced only by
the damage index d since the damage-
M- qb-pl test based model is uncoupled
-50
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1
rotations ,
pl

Progression of the Damage Functions with
60 the damage index d – The damage threshold
Threshold Energy g=G-R R shows an increasing exponential behavior
50
governed by the damage index d and the
exponent of the damage evolution law
40
r=0.90. The energy function G represents
R(d),G(d),g(d)

30
the amount of energy stored during the
loading process and it is a typical step
20 function. The constant energy intervals
correspond to the unloading branches since
10 the unloading process in the damage-based
model is elastic. The initial damage threshold
0 is Gcr=0.0019 [KNm] and it corresponds to
the formation of the first crack in the
0.7 member. The function g represents the
-10
0 0.1 0.2 0.3 0.4 0.5 0.6
Positive Damage index damage criterion adopted for the model,
g=G–R>0

0.7
Damage Index Evolution – The
0.6 progression of the damage index with the
load cycles is captured in this graph. The
0.5 damage index is recorded only for positive

bending at the bottom of the cantilever
0.4
column, where the plastic hinge forms.
The damage index starts to grow after the
0.3
first crack forms in the member, it
0.2
progresses slowly until the first yielding of
the member and after that the progression
0.1 of the damage becomes steeper until
failure. The maximum value of damage is
0 dmax=0.5426
0 200 400 600 800 1000 1200 1400 1600 1800 2000
No. of iterations

Ductility factors – The graph reports the ductility of the member for every load cycle. The cumulative ductility of the member is also estimated
to μcum=38.4. The displacement at first yielding is approximated to δy=0.0113 [m]

200
model test

100
Force (KN)

Force-displacement curves
0 for experimental test and
numerical simulation
Specimen JSCE-5

-100

-200
-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08
Displacement (m)
500
model test

250
Force (KN)

0 for experimental test and
numerical simulation
Specimen 9

-250

-500
-0.12 -0.06 0 0.06 0.12
Displacement (m)

300
model test
200

100
Force (KN)

for experimental test and
0 numerical simulation
Specimen OCR1
-100

-200

-300
-0.12 -0.06 0 0.06 0.12
Displacement (m)
200
model test

100
Force (KN)

for experimental test and
0 numerical simulation
Specimen 806040

-100

-200
-0.18 -0.12 -0.06 0 0.06 0.12 0.18 0.24
Displacement (m)

The applications of the model and the comparisons of analytical with experimental
results demonstrate the ability of the proposed beam element to describe several
complexities of the hysteretic behavior of structural members, such as softening and
stiffness degradation.
REMARKS ON MODEL PERFORMANCE

Unlike material response, in which the possibility to reproduce cyclic degradation and
predict failure is directly related to the ability of the material models to characterize
post-yield softening of concrete, buckling of reinforcing bars, fracture of confining
reinforcement, etc., in this damage-based element model the simulation of the
softening behavior as well as the stiffness degradation of the structural member, are a
direct result of the adopted yield surface and the ability of the flexibility matrix of the
damaged member implemented in the one component model, to trace this behavior.

The agreement with the experimental results is good, especially, if the envelope curve
is compared. The first cycles shows some discrepancies between analysis and
experiment, because the effect of the first cycle was not included in the analysis and
the difficulty of establishing the right initial conditions for the model. All subsequent
analytical cycles show good agreement with the experimental results.

The main difficulty in the use of the model consists in the selection of the calibration
parameters, and, even if all of them bear a direct relation to the physical properties of
the structural member, it is possible to arrive at a physically unreasonable hysteretic
behavior by injudicious selection.
The selection of the load step size was, governed by the desire to obtain a smooth
load-displacement relation for the presentation of the results.

The calibration parameters selection not only affects the accuracy of the constitutive
relation of the model, but has an impact on the numerical convergence
characteristics as well. In this example convergence was always achieved very

rapidly at each load step never requiring more than 10-15 iterations.

The damage-based model works as a one component element model in which the
response is path-dependent. This implies that at the end of every incrementation
step, the final force depends on how the strain increment is partitioned between the
number of iterations. Therefore, small changes in the iteration strategy, that is the
choice of the load step size, can sensitively alter the final value of the force, even if
the final displacement solution remains the same.

The likelihood of an abnormal behavior increases with the number of load cycles,
because of the monotonically increasing value of energy dissipation, which affects
the degradation parameter d.

Load control method applied provides good results when the member stiffness is
high, but fail to trace the nonlinear behavior near the ultimate strength of the
member and the post-peak response (Filippou and Issa, 1988).

The model becomes a standard elastoplastic model with linear kinematic and
isotropic hardening if damage remains constant. On the contrary, the damage
produces a softening effect. Thus, the achievement of an accurate numerical
prediction, results from the competition between the hardening due to plastic
deformations and the softening that is the consequence of damage.


The stiffness reduction of the model during unloading is not as pronounced as in the
experimental results: while the model exhibits a practically straight unloading
branch, the specimens display a gradual loss of stiffness during unloading. This can
be attributed to the following factors: (a) even though the model accounts for
stiffness loss between cycles, it cannot accommodate graduate stiffness reduction
during the unloading phase; thus, the unloading branch of the model is straight
until reaching the yield strength. This is also due to the fact that, since we are
dealing with hysteretic nonlinear behavior modeled in lumped plasticity, the
modification of the extension of the inelastic zone during reversals cannot be taken
into consideration. (b) the use of the same degradation parameter d under, both
positive and negative loading, fails to reproduce properly opening and closure of the
cracks.

Regression-based identification to provide guidelines for estimating
IDENTIFICATION OF MODEL PARAMETERS dy
IDENTIFICATION OF MODEL PARAMETERS dy the required model parameters
Regression analyses of the data set were performed to understand which properties
of the member affect the accumulation of damage during the loading process, and
with which intensity as well as the weighted choice of the calibration parameters dy
and r which do not bear a directed relation to properties of the structural member.

L s
d y  0.1ln  0.17 ln  0.44 ln  ln l  0.3
h db
L NSC
r  19.24  1.62  0.24  f c     0.06 f y  22.17
h
AND r
AND r

s
d y  0.12 ln  2.64 ln  1.65 ln sh  2.53
db
HSC s
r  0.18  54.53l  0.002 f y  2.28
db

Force-
IDENTIFICATION OF MODEL PARAMETERS dy
IDENTIFICATION OF MODEL PARAMETERS dy displacement
curves for
experimental
test and
numerical
simulation
with
calibrated
parameters
dy=0.1 and
r=5.6.
Specimen
JSCE-5
AND r
AND r

Force-
displacement
curves for
experimental
test and
numerical
simulation
with
calibrated
parameters
dy=0.04 and
r=2.96.
Specimen 9

Damage-based models for flexural response on RC elements

Damage-based models for flexural response on RC elements

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