portal of damage: a web-based finite element program for the

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PROOF
Portal 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:
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(a)
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doi:1

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ADE

Create an account in a server.

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(b) DMake use of a semi-graphic pre-processor to create an input file with a digitalized version of the structure.
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(c) Run a dynamic finite element program and monitor the state of the process.
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(d) EDownload or upload input and output files in text format.
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(e) TMake use of a graphic post-processor.
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Cq 2005 Published by Elsevier Ltd.

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EKeywords: Web-based program; Finite element analysis; Reinforced concrete; Lumped damage mechanics; Frames
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COR1. Introduction
Massive 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

www.elsevier.com/locate/advengsoft

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http://www.elsevier.com/locate/advengsoft

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Fig. 1. Cracked plate of thickness B subjected to a force P.

AD

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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.

M.E. Marante et al. / Advances in Engineering Software xx (xxxx) 1–13 3


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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.

ADES 941—20/1/2005—13:22—SWAPNA—132742—XML MODEL 5 – pp. 1–13

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.

AD



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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.


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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ORREC
2.6. 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.


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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FFig. 6. Main page of the portal of damage.

AD



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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


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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http://portalofdamage.ula.ve

http://portalofdamage.ula.ve

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


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.

AD



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CORREC
Two 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.


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


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 shows
the 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.


(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].

AD



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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


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 reach
convergence

(c)
Non-existence of mathematical solutions
ED 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.



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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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ORRECReferences
[1] Cipollina A, Lopez-Inojosa A, Florez-Lopez J. A simplified damage

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


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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portal of damage: a web-based finite element program for the

Documents