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PROOFPortal of damage: a web-based finite element program for the analysis
of framed structures subjected to overloads
Marıa E. Marantea,b,*, Lorena Suareza, Adriana Queroa, Jorge Redondoa, Betsy Veraa,Maylett Uzcateguia, Sebastian Delgadoc, Leandro R. Leona, Luis Nunezd, Julio Florez-Lopeza
aSchool of Engineering, University of Los Andes, Merida 5101, VenezuelabSchool of Civil Engineering, Lisandro Alvarado University, Barquisimeto, Venezuela
cSchool of Engineering, University of Zulia, Maracaibo, VenezueladSchool of Sciences, University of Los Andes, Merida 5101, Venezuela
Received 8 April 2004; revised 10 April 2004; accepted 12 June 2004
Abstract
This paper describes a web-based finite element program called portal of damage. The purpose of the program is the numerical simulation
of reinforced concrete framed structures, typically buildings, under earthquakes or others exceptional overloads. The program has so far only
one finite element based on lumped damage mechanics. This is a theory that combines fracture mechanics, damage mechanics and the
concept of plastic hinge. In the case of reinforced concrete frames, the main mechanism of deterioration is cracking of concrete. Cracking in a
frame element is lumped at the plastic hinges. Cracking evolution in the plastic hinge is assumed to follow a generalized form of the Griffith
criterion. The behavior of a plastic hinge with damage is described via the effective stress hypothesis, as used in continuum damage
mechanics. The portal that can be accessed using any commercial browser (explorer, netscape, etc.) allows to:
8788
(a)0965
doi:1
* C
Facu
Tel./
E
ADE
Create an account in a server.
89
(b) DMake use of a semi-graphic pre-processor to create an input file with a digitalized version of the structure.90
(c) Run a dynamic finite element program and monitor the state of the process.91
(d) EDownload or upload input and output files in text format.92
(e) TMake use of a graphic post-processor.93
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Cq 2005 Published by Elsevier Ltd.
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97
EKeywords: Web-based program; Finite element analysis; Reinforced concrete; Lumped damage mechanics; Frames
R 9899
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COR1. IntroductionMassive collapse of civil engineering structures due to
earthquakes occurs regularly around the world, more often
in third-world countries. Post-disasters reconnaissance
reports have shown that in many cases the failure is due
UN-9978/$ - see front matter q 2005 Published by Elsevier Ltd.
0.1016/j.advengsoft.2004.06.017
orresponding author. Address: Department of Structural Engineering,
ltad de Ingenieria, University of Los Andes, Merida 5101, Venezuela.
fax: C58 274 2402867.
-mail address: [email protected] (M.E. Marante).
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to deficiencies in the materials and or in the construction
details. Probably, there are hundreds of thousands of
structures in seismic areas everywhere that need retrofitting.
Thus, the identification of these structures is an urgent task.
But if ever done, this mission cannot be the job of a couple
R&D engineers of some big company. On the contrary, it
will probably be the work of an entire community of local
engineers in the countries in question. It is also likely that
this will be only a part-time activity for these engineers.
The identification of vulnerable structures can be carried
out in two steps: first, the recognition of physical defects in
Advances in Engineering Software xx (xxxx) 1–13
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Fig. 1. Cracked plate of thickness B subjected to a force P.
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UNCORREC
the structure by using visual screening tools and second, the
quantification of the potential impact of these defects. One
of the tools that can be used for the latter goal is the
numerical simulation of the behavior of structures under
seismic overloads. Taking into account the size, the number
and the complexity of this kind of structures (for instance
buildings), the models used for this simulation must be not
only realistic but also simple and effective. Those are
obviously contradictory requirements. One of the possible
compromises is represented by lumped damage mechanics.
Lumped damage mechanics is a general framework for
the modeling and the numerical simulation of the process of
deterioration and collapse of framed structures. This theory
combines fracture mechanics, damage mechanics, and the
concept of plastic hinge. Lumped damage mechanics has
been used to model the behavior of reinforced concrete
frames [1–4], and steel frames [5,6]. In the former case, the
main mechanism of damage is cracking of the concrete. In
lumped damage mechanics, cracking as well as plasticity
are concentrated in plastic hinges. New sets of hinge-related
variables are then introduced that can take values between
zero and one as the conventional continuum damage
variable. Expressions of the energy release rate of a plastic
hinge are then derived and damage evolution in the plastic
hinge is described by a generalized form of the Griffith
criterion. Lumped models allow for a simple and effective
representation of many civil engineering structures. How-
ever, any realistic model of the process of deterioration and
collapse of structures is always expensive in computational
terms.
Taking into account all the previous remarks, it seems
that coupling nonlinear finite element programs with
internet can be a very good tool for the diagnosis of
potentially dangerous structures. This paper presents a
prototype of this kind of program. It has been named ‘portal
of damage’. The system consists in a dynamic and nonlinear
finite element program based on lumped damage mechanics
that can be accessed via internet. The pre-processing, post-
processing and monitoring of the computations can be done
using common commercial browsers such as Explorer or
Netscape. With this kind of program, users do not need to
buy expensive annual licenses and hardware but pay only
for the time and resources consumed. If a systematic effort
of security assessment is decided by some public or private
organization, it will also minimize and centralize the
funding needed for the computational requirements of the
task.
This paper is organized as follows. In Section 2, the
fundamental concepts of lumped damage mechanics are
summarized. The portal is described in Section 3 and some
numerical examples carried out with portal are presented in
Section 4. The first and second examples are numerical
simulations of tests reported in the literature. These results
give an idea of the precision that can be expected with the
portal. The final example presents the numerical simulation
ES 941—20/1/2005—13:22—SWAPNA—132742—XML MODEL 5 – pp. 1–13
of the behavior of a frame designed after the current
Venezuelan code subjected to earthquake forces.
2. Lumped damage mechanics
ED PROOF
2.1. Fundamental concepts of fracture mechanics
In brittle materials, it is shown that the elastic stresses at
the tip of an ideal crack tend to infinite independently of the
level of external forces applied far from the crack. Thus,
none of the classic failure criteria based on stresses or strains
can be used to predict crack propagation. In a classic paper
[7] (as described in [8]), Griffith proposed a criterion of
crack propagation based on an energy balance. Griffith
postulated that crack propagation can occur only if sufficient
potential energy can be released to overcome the resistance
to crack extension. In mathematical terms this criterion,
denoted usually as the Griffith Criterion, can be written as:
G Z R (1)
where G is called the energy release rate and is obtained
after a structural analysis, and R is denoted the crack
resistance function and is assumed to be a material property.
It can be shown that in a solid subjected to a tensile force P,
the energy release rate is given by the following expression:
G ZKvU
vAZ
vU�
vAZ
P2
2B
dF
da(2)
where U is the potential energy cumulated in the solid, U* is
the complementary potential energy, A is the crack area, a
the crack length, B the specimen thickness and F is the
compliance defined as the load-point displacement per unit
load (see Fig. 1). The crack resistance might be a function of
the crack extension. The Griffith criterion establishes that
crack propagation cannot occur if this energy release rate is
lower than the crack resistance.
T
P
Fig. 2. (a) Planar frame (b) generalized strains of a frame member (c) generalized stresses of a frame member (d) lumped dissipation model.
Fig. 3. Cracking representation in lumped damage mechanics.
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UNCORREC
2.2. Flexibility matrix of a reinforced concrete frame
member with cracking
It can be seen that in order to describe crack propagation
in a reinforced concrete element, within the framework of
fracture mechanics, the potential energy or the flexibility of
the cracked element must be obtained. Consider a planar
frame as shown in Fig. 2a. A member between the nodes i
and j is isolated from the frame. The generalized stress and
strain matrices of the frame member are given, respect-
ively, by stZ(mi,mj,n) and 3tZ(fi,fj,d), where the terms
mi and mj represent the flexural moments at the ends of the
element, n is the axial force, fi and fj are the relative
rotations of the frame member with respect to the chord
and d is the chord elongation (see Fig. 2b and c).
Two major inelastic phenomena occur in a reinforced
concrete structure subjected to overloads: yield of the
reinforcement and concrete cracking. In order to model
these phenomena in large and complex framed structures,
the lumped dissipation model of a frame member is usually
adopted. This model consists in assuming that all inelastic
phenomena can be lumped at special locations called plastic
or inelastic hinges. Therefore a frame member is considered
as the assemblage of an elastic beam-column and two
inelastic hinges as shown in Fig. 2d.
The conventional theory of elastic–plastic frames is
obtained by the introduction of an internal variable that
will be denoted in this paper generalized plastic strains
3tpZ ðf
pi ;f
pj ; 0Þ, where the symbols f
pi yf
pj represent the
plastic rotations of the inelastic hinges at the ends i
and j.
Lumped damage mechanics is obtained by the introduc-
tion of a new set of hinge-related internal variables. This
variable, denoted damage, measures the crack density in the
element as indicated in Fig. 3. Thus, the damage matrix is
given by: DZ(di,dj), where di and dj are damage parameters
that can take values between zero (no cracking) and one
(total damage). They represent cracking as lumped in the
hinges i and j.
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In [2] the following elasticity law of a damaged
reinforced concrete frame member was proposed:
ROOF3 K3p Z FðDÞs;
FðDÞ Z
F011
1 Kdi
F012 0
F021
F022
1 Kdj
0
0 0 F033
2666664
3777775
(3)
ED where F is the flexibility matrix of a cracked frame member
and the terms F0ij represent the coefficients of the elastic
flexibility matrix as given in textbooks of structural
analysis.
It can be noticed that for damage values equal to zero, F
becomes the elastic flexibility matrix of the classic
structural analysis theory. If a damage variable tends to
one, the corresponding flexibility term tends to infinite (or
the stiffness tends to zero) and the inelastic hinge behaves as
the internal hinges of the classic frame analysis. It is
assumed that the damage parameters evolve continuously
from zero to one following the generalized Griffith criterion
that will be defined in Section 2.3. In this way, stiffness
degradation is represented by the model.
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2.3. Generalized Griffith criterion
The complementary strain energy of a cracked frame
member can be obtained from (3):
U� Z1
2stð3 K3pÞ Z
1
2stFðDÞs (4)
Then, the energy release rates of the plastic hinges are given
by:
Gi ZvU�
vdi
Z1
2st vFðDÞ
vdi
s Zm2
i F011
2ð1 KdiÞ2; Gj
Zm2
j F022
2ð1 KdjÞ2
(5)
Therefore a generalized Griffith criterion for the inelastic
hinge i can be defined in the following terms: there will be
damage evolution (i.e. crack propagation) in the plastic
hinge i only if the energy release rate Gi reaches the value of
the crack resistance of the hinge:
_di Z 0 if Gi KRðdiÞ!0 or _Gi K _RðdiÞ!0
_diO0 if Gi KRðdiÞ Z 0 and _Gi K _RðdiÞ Z 0
((6)
where R is the crack resistance of the plastic hinge i. This
function has been identified from experimental results [1]:
RðdiÞ Z Gcr Cqlogð1 KdiÞ
1 Kdi
(7)
where Gcr and q are member dependent parameters.
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UNCORREC2.4. Plastic behavior of a frame member with cracking
Following [2], the behavior of a plastic hinge with
cracking can be obtained by using the equivalent stress
hypothesis. In continuum damage mechanics, this hypoth-
esis states that the behavior of a damage material can be
expressed by the same relations of the intact material if the
stress is substituted by the effective stress. By analogy with
continuum damage mechanics, the effective moment �mi on a
plastic hinge is defined as:
�mi Zmi
1 Kdi
(8)
Thus, the yield function of a plastic hinge with damage and
linear kinematic hardening is given by:
fi Zmi
1 Kdi
Kcfpi
��������Kky%0 (9)
where c and ky are member-dependent properties. The
elasticity law (3), the Griffith criterion (6) and the yield
function (9) define the constitutive law of a frame member
with cracking and yielding.
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ED PROOF
2.5. Influence of the axial load
The coefficients ky, c, Gcr and q for hinges i and j can be
computed by the resolution of a system of nonlinear
equations. Consider a monotonic loading on a reinforced
concrete member. When the moment on the hinge reaches
the cross-section cracking moment Mcr, the energy release
rate is, for the first time, equal to the crack resistance:
mi Z Mcr0Gi Z Rð0Þ; therefore Gcr ZM2
crF011
2(10)
First cracking moment can be computed by using classic
theory of reinforced concrete. It is well known that the
parameter Mcr depends on the values of the axial force n on
the hinge. Therefore, Gcr is also a function of the axial force.
For higher values of the moment, damage evolution
starts; Griffith criterion defines then a relationship between
these two variables:
m2i F0
11
2Z ð1 KdiÞ
2Gcr Kqð1 KdiÞlnð1 KdiÞ (11)
According to (11), the moment is equal to Mcr for di equal to
0; then increases for higher values of damage; reaches a
maximum denoted Mu, for di equal to du and finally is equal
to zero for di equal to one. The value of the ultimate moment
Mu of the cross section can also be determined by the
conventional reinforced concrete theory and is again a
function of the axial force on the hinge. The search of
critical values of m2i leads to the following equation:
K2ð1 KduÞGcr Cq½lnð1 KduÞC1� Z 0 (12)
which allows for the computation of du. Thus, the parameter
q can be obtained by the resolution of the following
equation:
M2uF0
11
2Z ð1 KduÞ
2Gcr Kqð1 KduÞlnð1 KduÞ (13)
The last member-dependent coefficients are those of the
yield function: ky, c. When the moment on the hinge reaches
the value of the yield moment Mp, the corresponding
damage value dp can also be computed by the Griffith
criterion:
M2pF0
11
2ð1 KdpÞKGcr Kq
lnð1 KdpÞ
1 Kdp
Z 0 (14)
Additionally, for this value the yield function is for the first
time equal to zero and the plastic rotation is still nil,
therefore:
Mp K ð1 KdpÞky Z 0 (15)
Expression (15) allows for the determination of the
coefficient ky. The yield moment is again a function of the
axial force and can be computed by the classic theory of
reinforced concrete.
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Finally, the value of the plastic rotation Fpu that
corresponds to the ultimate moment Mu can be used to the
computation of the last parameter of the model c. i.e. the
expression fZ0 leads to:
Mu K ð1 KduÞðcFpu CkyÞ Z 0 (16)
It is assumed that interaction diagrams of Mcr, Mp, Mu and
Fpu can be computed with reasonable precision for cross
sections of any shape and reinforcement. Specifically it is
assumed that the ultimate plastic rotation can be estimated
as:
Fpu ycu
plp (17)
where cup is the ultimate plastic curvature and lp an
equivalent plastic hinge length.
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ORREC2.6. Other lumped damage mechanics concepts
Further extensions of the model described in the
precedent sections are possible and were implemented in
the portal. The most important of them is the concept of
unilateral damage. In frame members subjected to loadings
that change sign, two different set of cracks appears in the
frame member, one due to positive moments (positive
cracks) and another as a consequence of negative moments
(negative cracks). Within the framework of the lumped
damage mechanics, this effect can be represented by using
two sets of damage variables instead of one: DCZ ðdCi ; d
Cj Þ
and DKZ ðdKi ; d
Kj Þ, where the superscript indicates damage
due to positive or negative moments (see Fig. 4). A frame
element under this condition presents a behavior that can be
described as ‘unilateral’. This adjective, which comes from
continuum damage mechanics, indicates that for positive
moments, the negative cracks tend to close and have little
influence on the member behavior and vice versa. A perfect
unilateral behavior can be described by the following
modification of the elasticity law (3):
3 K3p Z FðDCÞhsiC CFðDKÞhsiK (18)
where the symbols hxiC and hxiK indicate, respectively, the
positive and negative part of the variable x. i.e. hxiCZx if
xO0; hxiCZ0 otherwise. hxiKZx if x!0; hxiKZ0 other-
wise. It can be noticed that positive damage has no influence
at all in the compliance of the structure if the moments are
negative and vice versa.
UNC
Fig. 4. Unilateral damage.
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Two different energy release rates per hinge are now
defined: GCi and GK
i . They are given by an expression
similar to (5) except that the moment is substituted by its
positive or the negative part and the damage for the
corresponding positive or negative variable. The yield
function can also be modified by the definition of an
effective moment with the positive damage variable if
positive cracks are open and vice versa (see [2], for
additional details).
Another extension of the model consists in the use of
modified forms of the Griffith criterion. In particular the one
described in [9], was included in the portal in order to
represent low cycle fatigue effects.
Finally, although models for steel structures [5,6] and
three-dimensional reinforced concrete frames [4] have been
developed and tested, they have not been implemented in
the portal yet.
ED PROOF2.7. A finite element based on lumped damage mechanics
A finite element can be described as a set of equations
that relates the element degrees of freedom utZ(ui,vi,qi,uj,
vj,qj) with the nodal forces Q (see Fig. 5). The member
strain matrix and the degrees of freedom of the element are
related by the following kinematic equations:
_fi Zsin aij
Lij
_ui Kcos aij
Lij
_vi C _qj Ksin aij
Lij
_uj Ccos aij
Lij
_vj
_fj Zsin aij
Lij
_ui Kcos aij
Lij
_vi Ksin aij
Lij
_uj Ccos aij
Lij
_vj C _qj
_d ZK_ui cos aij K _vi sin aij C _uj cos aij K _vj sin aij
;
i:e: _3 ZB _u
(19)
where B is the transformation matrix. If geometrically
nonlinear effects are taken into account, the angle aij and the
length Lij are not constant but depend on the displacements
u. Geometrically nonlinear effects are included in the portal
in this way.
Fig. 5. Degrees of freedom of a frame element and nodal forces.
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T
FFig. 6. Main page of the portal of damage.
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The nodal forces and the stress matrix are also related by
the following expression:
Q Z Bts (20)
The set of equations composed by (19,20) and the
constitutive law described in the previous sections define
a finite element that can be included in the library of
elements of commercial nonlinear analysis programs [3].
Additionally, a new finite element program that can be
accessed via web has also been developed and is described
in Section 3.
UNCORREC
Fig. 7. Diagrams of interaction com
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ED PROO
3. Description of the portal
3.1. Links of the portal
The portal can be accessed in the address http://
portalofdamage.ula.ve. The system has a user’s data base,
thus she/he has to register in her/his first visit. After
registration, the user has access to the five links of the
system: pre-processor, processor, post-processor, user
manual and theory manual (see Fig. 6).
Within the pre-processor (a Java program), the user has
different menus for the description of the frame geometry,
the boundary conditions, the dimensions of the elements’
puted by the pre-processor.
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T
Fig. 8. Monitoring of an analysis in the portal.
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cross sections (only rectangular cross sections are included
so far), the amount and location of the longitudinal and
transversal reinforcement, the uniaxial behavior of the
concrete and the reinforcement and, finally, the loading. The
user can impose distributed forces on the elements, and
displacements, accelerations or forces on the nodes.
However, as indicated in Section 3, the model needs
interaction diagrams of the first cracking moment,
yielding moment, ultimate moment and curvature. All
these diagrams are computed by a FORTRAN program
embedded into the pre-processor called diagram gen-
erator. The diagram generator uses as input the uniaxial
behavior of the materials, the dimensions of the cross
section and the amount of the reinforcement. This
process is carried out by using standard methods of the
classic reinforced concrete theory for confined (Kent and
Park model) or non-confined elements (Hognestad
UNCORREC
Fig. 9. Graphic po
ADES 941—20/1/2005—13:22—SWAPNA—132742—XML MODEL 5 – pp. 1–13
ED PROOFmodel). The details of these procedures can be seen in
any textbook on reinforced concrete, for instance [10].
The user instructs the portal to perform the generation of
the interaction diagrams in one of the menus of the pre-
processor. The results of these computations are shown,
graphically, by the portal (see Fig. 7). Notice that a table
with the coordinates of the diagrams is also created. The
user can modify these values. This is useful for some
special applications. Errors up to 15% in the computed
diagrams can be expected by using standard reinforced
concrete theory. In the case of very important structures,
it can be convenient the execution of experimental tests
on some typical elements of the frame. The experimental
results can be inserted directly in the analysis by the
modification of the diagram tables. Numerical results for
single elements of other programs or procedures can also
be used in this same way.
st-processor.
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FFig. 10. (a) Moment vs. rotation in a cantilever specimen after [12] (b) numerical simulation.
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CORRECTwo different kinds of data files are generated by the pre-
processor. The first one has the raw data of the frame
including the uniaxial behavior of the materials and details
of the cross section. A second data file is also generated;
it has the values of the interaction diagrams of the model
parameters instead of the element details. Only the second
kind of data files can be processed by the FE program.
The second link of the portal gives access to a page called
processor. In this page, the user may select a data file in
her/his account and process it with the FE program. The user
may also abort the analysis or monitor the process by opening
or updating a file with the relevant information (see Fig. 8).
When the process is successfully finished, the user may
download a results file in text format or may proceed to a
graphic evaluation of these results with the third main link
of the portal: the post-processor.
With the post-processor (see Fig. 9), the user may plot
histories of element variables (damage, generalized stresses
or strains, plastic rotations) or node variables (forces,
displacements, velocities, accelerations) and variable
vs. variable graphs. Damage distribution over the frame at
any instant of the loading can also be plotted.
3.2. The finite element program
All the previously described elements of the portal (pre-
processor, processor and post-processor) are Java programs.
UNTable 1
Data for the simulation presented in Fig. 10
F 0c (kg/cm2) e0 euc E (kg/cm2) eccu esm
468.05 0.002 0.004 200.000 0.005 0.125
Note: kg/cm2Z0.0980665 MPa.
ES 941—20/1/2005—13:22—SWAPNA—132742—XML MODEL 5 – pp. 1–13
ED PROOIn particular, the processor is a Java interface with a Fortran
FE program. This program manages the step by step
procedure and solves the dynamic equilibrium Eq. (21) at a
given instant after time discretization by using the Newmark
method:
LðUÞ ZXm
bZ1
QðUÞb CIðUÞKP Z 0 (21)
where I represents the inertia forces, U is the matrix of nodal
displacements of the entire frame, P is the external forces
and m represents the number of frame members. The
nonlinear problem (21) is solved by a standard Newton
method. For each iteration of the global problem (21), m
local problems have to be solved. Each local problem
consists on the computation of the nodal forces Q from the
nodal displacements U by using the constitutive law and
kinematic equations described in Section 2. The local
problem is also nonlinear and is solved by the Newton
method too. In fact, the local problem can become highly
nonlinear for high values of damage (over 0.5). However, in
a frame only few elements reach such high values of
damage. As a result, the convergence requirements of the m
local and the global problems are very diverse. Thus, a sub-
stepping strategy has been adopted: the local problem in
those elements, where damage is concentrated is solved by
ey esh fsu (kg/cm2) fy (kg/cm2) fyh (kg/cm2)
0.002 0.006 6038.7 4313.4 4313.4
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Fig. 11. Data for the description of the uniaxial behavior. (a) Concrete (b)
reinforcement.
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the discretization of the global time step in smaller local
steps (see [4]).
It can be noticed that lumped damage laws are
multicriteria models. Therefore special kinds of elastic
predictor–inelastic corrector algorithms are needed. In
particular, the one studied in [11] was used in the program.
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4. Numerical examples
Some numerical simulations carried out with the portal
are shown in this section. The goal of the two first examples
is to show the precision that can be expected by using the
portal. These examples correspond to the numerical
UNCORRECT
Fig. 12. (a) Moment vs. rotation in a cantilever sp
ADES 941—20/1/2005—13:22—SWAPNA—132742—XML MODEL 5 – pp. 1–13
ED PROOF
simulation of tests reported in the literature. The last case
is designed to show what could be a real application of
the portal: the numerical simulation of a building designed
after the current Venezuelan code.
The first example is a RC column in cantilever subjected
in laboratory to a complex loading program that represents
the actions that a building column must withstand during an
earthquake. They include cyclic lateral displacements and
variable axial forces [12]. The experimental behavior is
represented in the graph of lateral displacements vs. lateral
forces shown in Fig. 10a. The numerical simulation
obtained with the portal is shown in Fig. 10b. The data
used for the simulation is shown in Table 1. The meaning of
the symbols used in Table 1 is shown in Fig. 11.
It can be noticed, that although the results are very good
qualitatively, the portal underestimates the strength of the
column. This is a consequence of using classic reinforced
concrete theory.
A test like this one could also be part of the assessment
process of an entire structure. For instance, the specimen
could be a replica of one column of the structure that must
be evaluated. The diagrams generated by the portal can be
modified to fit the experimental results in order to carry out a
more precise simulation of the structure as mentioned in
Section 3. The numerical simulation of the same test carried
out with modified diagrams is shown in Fig. 12b. The
original and modified diagrams, i.e. the ones used in,
respectively, the simulations shown in Figs. 10b and 12b are
indicated in Fig. 13.
The second example is a two-story RC frame that was
subjected in laboratory to constant axial forces and to
imposed lateral displacement at the top of the frame as
shown in Fig. 14 (see [13]). The experimental results are
shown in Fig. 15a in a graph of lateral displacement
vs. lateral force. The results of the numerical simulation
ecimen after [12] (b) numerical simulation.
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UNCORRECTED PROOF
Fig. 13. (a) Interaction diagrams used in the simulation of Fig. 10b. (b) Interaction diagrams used in the simulation of Fig. 12b.
Fig. 14. Push over test on a RC frame after [13].
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TED PROOF
Fig. 16. Damage distribution at the end of the test.
Fig. 15. (a) Force vs. displacement in a frame after [13] (b) numerical simulation.
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CORRECwith the portal are shown in Figs. 15b and 16. Fig. 16 showsthe damage distribution at the end of the simulation. The
simulation was carried out using the diagrams of the portal
without modifications. The data used for the generation of
the diagrams is given in Table 2.
This damage distribution can be used to establish the
vulnerability of the structure as well as the feasibility of
reparation as discussed in [14]. For instance, values of
damage less than 0.2 can be described as minor, repairable
damages are in the range [0.2, 0.45] and values over 0.6
corresponds to progressive collapse. Theoretical and exper-
imental justification for these values can be found in [14].
The last example shows a simulation of what could be a
real application of the portal. A 10-story frame [15]
UNTable 2
Data for the simulation presented in Figs. 15b and 16
F 0c (kg/cm2) e0 euc E (kg/cm2) eccu esm
480.00 0.0018 0.004 200.000 0.0054 0.11
Note: kg/cm2Z0.0980665 MPa.
ADES 941—20/1/2005—13:22—SWAPNA—132742—XML MODEL 5 – pp. 1–13
(see Fig. 17) designed according to the current Venezuelan
code, which is strongly inspired on the USA one, was
subjected to two different acceleration records. The first one
was artificially generated from the spectrum of the design
earthquake that corresponds to medium seismic risk in
Venezuela. The damage distribution at the end of the
simulation is shown in Fig. 18. The maximum damage
observed is 0.37, which corresponds to repairable damage,
and a scattered pattern of damage in beams and columns can
be observed. These results indicate an adequate design of
the frame for this loading. The second one was also
artificially generated, but this time from the spectrum of
a design earthquake for the most unfavorable region
considered in the Venezuelan code. The intention of this
ey esh fsu (kg/cm2) fy (kg/cm2) fyh (kg/cm2)
0.0022 0.0095 6077.41 4262.43 4629.53
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T
Fig. 17. Ten-story RC frame [15].
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example is to evaluate the behavior of an under-designed
building. The damage distribution at the end of the second
simulation is shown in Fig. 19. The maximum damage is
this time 0.69 and other high values can be observed in
the middle zone of the frame. Such high values of damage
indicate that the frame was on the verge of structural
collapse. As a conclusion, a building like this one built in a
zone, where such an overload is likely to occur deserves
special attention.
UNCORREC
Fig. 18. Damage distribution at the end of th
ES 941—20/1/2005—13:22—SWAPNA—132742—XML MODEL 5 – pp. 1–13
5. Final remarks
This paper describes a web-based nonlinear, dynamic
finite element program. The authors do not have other
references to compare with in the field of structural and
solid mechanics. Beside the arguments mentioned in the
introduction of this paper, web-based FE programs also
exhibit obvious advantages from the point of view of the
author’s rights protection.
It must be underlined that the system is just a prototype
and that the debugging capacity of the team that developed
the portal is limited. The portal has been tested on a
restricted number of examples, but there may be some bugs
hidden in the software. The FE program is not yet very
efficient and computational times can be significantly
reduced. An effort of optimization and parallelization of
the FE program has already been initiated. There are
convergence problems in the case of severe overloads that
lead to very high values of damage. There may be several
reasons for these problems:
(a)
e ana
FUnknown bugs in FE program
(b)
OPossible failures of the chosen Newton method to reachconvergence
(c)
Non-existence of mathematical solutionsED PRO
It is important to clarify the last potential lack of
convergence cause. Any realistic model of the process of
damage and collapse of structures under mechanical over-
loads must include that possibility. In fact, structural collapse
can be mathematically defined in that way. However, so far
there is no criterion that establishes in the most general case
the existence of mathematical solutions to the lumped
damage problem. Some efforts have been undertaken in
that direction [16] but this is still an open problem. The goal
would be the development of uniqueness and existence
lysis with the design earthquake.
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Fig. 19. Damage distribution at the end of the analysis with a stronger earthquake.
M.E. Marante et al. / Advances in Engineering Software xx (xxxx) 1–13 13
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criteria that could be numerically evaluated by the portal and
predict in this way the structural collapse of the frame. In the
absence of such criterion, it is not possible to discard the first
or second cause of non-convergence if the portal fails to
finish a particular application.
The use of the portal is free and will remain free for
academic purposes within the limits of the resources
allocated to it.
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Acknowledgements
The results presented in this paper were obtained in the
course of an investigation sponsored by FONACIT.
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mechanics approach to nonlinear analysis of frames. Comput Struct
1995;54(6):1113–26.
[2] Florez-Lopez J. Frame analysis and continuum damage mechanics.
J Eur Mech 1998;17(2):269–84.
[3] Perdomo ME, Ramirez A, Florez-Lopez J. Simulation of damage in
RC frames with variable axial forces. Earthquake Eng Struct Dyn
1999;28(3):311–28.
[4] Marante ME, Florez-Lopez J. Three-dimensional analysis of
reinforced concrete frames based on lumped damage mechanics. Int
J Solids Struct 2003;40(19):5109–23.
UNC
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ED PROOF[5] Inglessis P, Medina S, Lopez A, Febres R, Florez-Lopez J. Modeling
of local buckling in tubular steel frames by using plastic hinges with
damage. Steel Compos Struct 2002;2(1):21–34.
[6] Febres R, Inglessis P, Florez-Lopez J. Modeling of local buckling in
tubular steel frames subjected to cyclic loading. Comput Struct 2003;
81:2237–47.
[7] Griffith AA. The phenomenon of rupture and flow in solids. Phil Trans
R Soc Lond 1921;A221:163–97.
[8] Broek D. Elementary engineering fracture mechanics. Dordrecht:
Martinus Nijhoff Publishers; 1986.
[9] Thomson E, Bendito A, Florez-Lopez J. Simplified model of low
cycle fatigue for RC frames. J Struct Eng ASCE 1998;124(9):
1082–6.
[10] Park R, Paulay T. Reinforced concrete structures. New York: Wiley;
1975.
[11] Simo J, Kennedy A, Govindjee S. Non-smooth multisurface
plasticity and viscoelasticity, Loading/unloading conditions and
numerical algorithms. Int J Numer Methods Eng 1988;26:
2161–85.
[12] Abrams DP. Influence of axial force variations on flexural behavior of
reinforced concrete columns. ACI Struct J 1987;May–June.
[13] Vechio FJ, Emara MB. Shear deformation in reinforced concrete
frames. ACI Struct J 1992;89(1).
[14] Alarcon E, Recuero A, Perera R, Lopez C, Gutierrez JP, De Diego A,
et al. A reparability index for reinforced concrete members based on
fracture mechanics. Eng Struct 2001;23(6):687–97.
[15] Pernia F, Estudio parametrico del analisis de push over aplicado a
edificios de concreto armado, Thesis, University of los Andes, Merida
(in progress).
[16] Marante ME, Picon R, Florez-Lopez J, Analysis of localization
in frame members with plastic hinges, Int J Solids Struct in
press.
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