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Erratum

Online publication date: 08 July 2010

To cite this Article (2010) 'Erratum', Journal of Earthquake Engineering, 14: 6, 953To link to this Article: DOI: 10.1080/13632469.2010.495681URL: http://dx.doi.org/10.1080/13632469.2010.495681

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Journal of Earthquake Engineering, 14:953, 2010

Copyright � A.S. Elnashai & N.N. Ambraseys

ISSN: 1363-2469 print / 1559-808X online

DOI: 10.1080/13632469.2010.495681

Erratum

In volume 14, supplement 1 (2010), the article, ‘‘Numerical Issues in Distributed

Inelasticity Modeling of RC Frame Elements for Seismic Analysis’’ by A. Calabrese

et al. was published with errors. The entire article is reprinted here using the same page

numbers as the originally printed article. The corrected article can be found online in

volume 14, supplement 1 at www.informaworld.com/UEQE. Any citations of the article

should be as follows:

Calabrese, A., Almeida, J.P., Pinho, R. 2010. Numerical Issues in Distributed Inelasticity

Modeling of RC Frame Elements for Seismic Analysis. Journal of Earthquake

Engineering 14(S1), 38-68.

953

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Journal of Earthquake Engineering, 14(S1):38–68, 2010

Copyright � A.S. Elnashai & N.N. Ambraseys

ISSN: 1363-2469 print / 1559-808X online

DOI: 10.1080/13632461003651869

Numerical Issues in Distributed InelasticityModeling of RC Frame Elements for

Seismic Analysis

ARMANDO CALABRESE1, JOAO PACHECO ALMEIDA1,and RUI PINHO2

1ROSE School, IUSS Pavia, Pavia, Italy2Dipartimento di Meccanica Strutturale, Universita di Pavia, Pavia, Italy

The sustained development of computational power continues to promote the use of distributedinelasticity fiber frame models. The current article presents a comprehensive application and discus-sion of state-of-the-art formulations for the nonlinear material response of reinforced concretestructures. The broad character of the study is imparted by the joint analysis of the effects of thetype of formulation (displacement-based versus force-based), sectional constitutive behavior (hard-ening versus softening response), and numerical integration parameters (such as quadrature method,mesh definition, or number of integration points). Global and local responses are assessed, along witha critical review of existing regularization techniques. An experimentally tested cantilever is used toconduct the study and illustrate the previous features. The example shows that the convergence ofdisplacement-based meshes under objective response can be much slower than what preceding studiesindicate, unlike their force-based counterpart. Additionally, the physical interpretation of the localresponse under softening behavior supports the proposal of a novel regularization scheme fordisplacement-based elements, validated through comparison against experimental results.

Keywords Reinforced Concrete (RC); Finite Elements (FE); Fiber Elements; Displacement-Based Elements (DB); Force-Based Elements (FB); Localization; Regularization

1. Introduction

Distributed inelasticity elements are becoming widely employed in earthquake engineer-

ing applications, either for research or professional engineering purposes. Their advan-

tages in relation to the simpler lumped-plasticity models, together with a concise

description of their historical evolution and discussion of existing limitations, can be

found in Fragiadakis and Papadrakakis [2008]. In particular, it is noted that distributed

inelasticity elements (1) allow the inelastic behavior to spread throughout the entire

element and (2) do not require any calibration of their parameters against the response

of an actual or ideal frame element under idealized loading conditions.

The present study uses the renowned fiber approach to represent the cross section

behavior, where each fiber is associated with a uniaxial stress–strain relationship. Such

models feature additional assets, which can be summarized as follows: no requirement of

a prior moment-curvature analysis of members; no need to introduce any element

hysteretic response (as it is implicitly defined by the material constitutive models); direct

modeling of axial load-bending moment interaction (both on strength and stiffness);

Received 18 September 2009; accepted 20 January 2010.

Address correspondence to Armando Calabrese, ROSE School, c/o EUCENTRE, Via Ferrata 1, 27100

Pavia, Italy; E-mail: acalabrese@roseschool.it

38

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straightforward representation of biaxial loading; and interaction between flexural

strength in orthogonal directions.

Distributed inelasticity frame elements can be implemented with two main different

finite elements (FE) formulations: the classical displacement-based (DB) ones [e.g.,

Hellesland and Scordelis, 1981; Mari and Scordelis, 1984] and the more recent force-

based (FB) formulations [e.g., Neuenhofer and Filippou, 1997; Spacone et al., 1996].

Mixed formulations have also been developed [e.g., Alemdar and White, 2005]. In a DB

approach the displacement field is imposed, whereas in an FB element equilibrium is

strictly satisfied and no restraints are placed on the development of inelastic deformations

throughout the member. For this reason, FB formulations are extremely appealing for

earthquake engineering applications, where significant material nonlinearities are expected

to occur. From the theoretical viewpoint, the differences between these approaches are well

known [e.g., Alemdar and White, 2005; Hjelmstad and Taciroglu, 2005].

For a hardening type of sectional behavior, both FB and DB elements produce an

objective response at the global (force-displacement) and local (moment-curvature)

levels, whereas the results are nonobjective in the case of a softening sectional law.

This numerical issue, commonly known as strain localization or simply localization, was

firstly discussed by Zeris and Mahin [1988] for DB elements, while Coleman and Spacone

[2001a] studied it for FB elements. It is recalled that, particularly in the case of reinforced

concrete (RC) frame elements, the increase of the axial load ratio can induce the change

between a fully hardening and a softening type of sectional constitutive behavior.

To restore the response objectivity for DB elements, it is usual to define the length of

the mesh extremity elements as the plastic hinge length [e.g., Coleman and Spacone,

2001b; Legeron and Paultre, 2005; Papaioannou et al., 2005]. On the other hand, the main

regularization procedures for FB elements have been proposed by Coleman and Spacone

[2001a], Scott and Fenves [2006] and Addessi and Ciampi [2007]. More recent develop-

ments in the regularization of FB elements have also been suggested by Scott and

Hamutcuoglu [2008] and Lee and Filippou [2009].

Besides the aforementioned studies, significant works are the one by Neuenhofer and

Filippou [1997] regarding the identification of the distinct convergence characteristics of the

DB and FB formulations for hardening systems and the study by Papaioannou et al. [2005],

which placed a particular emphasis on the analysis of the numerical integration scheme.

In contrast with the preceding clear theoretical setting, and to the authors’ knowl-

edge, there is no published study that jointly tackles, from the application viewpoint

concerning structural engineering practice, all the parameters that influence the perfor-

mance of such models: type of formulation (DB versus FB), sectional constitutive

behavior (hardening versus softening response), numerical integration scheme (Gauss-

Legendre versus Gauss-Lobatto), and mesh characteristics (number of elements per

member and number of integration points per element). Additionally, the purpose of

the present article includes the inspection, from a critical viewpoint based on the physical

interpretation of the localization phenomenon, of the main regularization techniques.

Again, this was thought to be of relevance from the structural engineering application

standpoint. Both the global and local-level responses were taken into account.

The following section briefly addresses the main FE models commonly used for

inelastic analysis of frame elements, focusing in particular on the added merits of

complementing such formulations with the traditional sectional fiber approach.

An inspiring parallelism between the physical occurrence of localized responses and the

numerical counterpart of such phenomenon is then carried out, illustrating the importance of

establishing a connection between the fracture mechanics framework and the FE context.

Numerical Issues in Distributed Inelasticity 39

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Several analyses on a literature case study are then performed according to the

previously described objective. Besides the expected amount of useful information,

some novel aspects deserving special consideration arise from the present work.

In particular, it is shown to be of interest to deepen the comparison between the

convergence of the objective hardening responses of DB versus FB formulations. Although

the delayed convergence of the former type of models was noticed by Neuenhofer and

Filippou [1997], the current article shows not only that the relevance of this issue can be

larger than initially expected but also addresses the corresponding justification.

Finally, the critical interpretation of the local responses indicates that physically

meaningful numerical results call for the introduction of a new regularization proposal for

DB elements. The comparison against the experimental tests of the present cantilever

validates this method for restoring objectivity.

2. FE Models for Inelastic Analyses of Frame Structures

Two main general approaches are used in order to perform numerical analyses of frame

structures accounting for material nonlinearity: lumped (or concentrated) inelasticity and

distributed inelasticity (which includes fiber models). Chronologically, lumped plasticity

formulations have appeared earlier, being used for earthquake engineering purposes since

the 1960s. Nowadays, especially due to increasing computational power, distributed inelas-

ticity models are also gradually becoming more popular in standard professional practice.

2.1. Distributed Inelasticity Models

In this approach the entire member is modeled as an inelastic element, the source of such

inelasticity being defined at the sectional level. The global inelasticity of the frame is then

obtained by integration of the contribution provided by each controlling section (Fig. 1a).

A major advantage of such models is the nonexistence of a predetermined length where the

inelasticity can occur since all the sections can have excursions in this field of response.

Though this approach is a closer approximation to reality, it also requires more computa-

tional capacity; that is, more analysis time, as well as memory and disk space.

A very common and useful way of computing the sectional response is via a refined

discretization into relatively small domains (Fig. 1b), which follow a material uniaxial

inelastic behavior.

Cover concrete (unconfined)

Core concrete (confined)

Steel bars

z

y

x

y

z

Integration sections

FIGURE 1 (a) Example of controlling sections along the element and (b) section fiber

discretization.

40 A. Calabrese, J. P. Almeida, and R. Pinho

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Such an approach, commonly referred to as fiber modeling, has an inarguable asset: no

previous calibration of the moment-curvature hysteretic rule is required; therefore, there is no

need for any semi-empirical decision. On the other hand, it is difficult to reproduce the cyclic

flexural-shear interaction and degradation, which is a current subject of wide research.

Fiber elements are commonly modeled with two different FE formulations. The DB

formulation is based on the element stiffness. On the other hand, the FB formulation is built

on the element flexibility. In a DB element the displacement field is imposed and the element

forces are found by energy considerations. Conversely, in an FB element the force field is

imposed, with the element displacements being obtained by work equivalence balance.

In the DB case, displacement shape functions are used, corresponding, for instance, to a

linear variation of curvature along the element. In contrast, in an FB approach, a linear moment

variation is imposed; i.e., the dual of the previously referred linear variation of curvature. For

linear elastic material behavior, the two approaches obviously produce the same results,

provided that only nodal forces act on the element. On the contrary, in case of material

inelasticity, imposing a displacement field does not enable capturing the real deformed shape

because the curvature field can be, in a general case, highly nonlinear. In this situation, with a

DB formulation a refined discretization (meshing) of the structural member is required, in order

to accept the assumption of a linear curvature field inside each of the subdomains.

Instead, an FB formulation is always exact, because it does not depend on the

assumed sectional constitutive behavior. In fact, it does not restrain in any way the

displacement field of the element. In this sense this formulation can be regarded as

exact, with the only approximation being introduced by the discrete number of the

controlling sections along the element that are used for the numerical integration. Such

an interesting feature enables modeling each structural member with a single FE, there-

fore allowing a one-to-one correspondence between structural members (beams and

columns) and model mesh elements. In other words, no meshing is theoretically required

within each element, even if the cross section is not constant. This is because the force

field is always exact, regardless of the level of inelasticity.

Further, it is immediate with FB formulations to take into account loads acting along

the member, whereas this is not the case for DB approaches. Indeed, DB elements need to

be subdivided in order to approximate the curvature distribution induced by distributed

loads along the frame element, whereas FB elements do not require discretization even

when subjected to span loads.

3. Localization Issues

3.1. Localization From a Physical Viewpoint

In the research of fracture mechanics of concrete, the concepts of size effect and

localization are very well known and have been studied for several years. For this reason,

there is a large amount of literature focusing on this subject [e.g., Markeset and

Hillerborg, 1995; Weiss et al., 2001]. In this context, the term localization indicates the

experimental evidence that the descending stress–strain curve becomes specimen size

dependent and, consequently, it cannot be considered as a pure material property. It has also

been established that strain softening is often spread over a finite-size region of the

material. The above findings mean that a concrete specimen tested in compression will

be damaged, and eventually ‘‘collapse’’, due to a local mechanism, caused by the concen-

tration of strains into a limited (localized) region of the whole body. As a result, the global

response, namely the stress–strain curve, does not depend uniquely on concrete properties

but on the specimen size (size effect), regardless of the adopted test procedure; see Fig. 2.

Numerical Issues in Distributed Inelasticity 41

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As the sample becomes longer, the total post-peak response of the whole specimen

becomes steeper or more brittle, leading to the possible occurrence of severe snapbacks,

particularly in the testing of high-strength concrete. It should be noted, however, that

localization was first observed for tensile tests, where it is more immediate to think about a

localized deformation (a crack), and only later the concept has been extended to compression.

Hence, these two physical phenomena, localization and size effect, exist both in tension and

in compression. Several models have been proposed by different authors to describe this

complex behavior [Bazant, 1989; Bazant et al., 1984; Hillerborg, 1990, among others].

3.2. Localization From a Numerical Viewpoint

In the previous paragraph, the definition of localization in the framework of fracture

mechanics has been presented. The same term is also frequently used in computational

mechanics to indicate an analogous FE pathology that occurs under the same physical

conditions; i.e., softening constitutive behavior. Zeris and Mahin [1988] indicated that a

conventional DB formulation is unable to establish solutions associated with softening

behavior. Coleman and Spacone [2001a] comprehensively addressed the problem in an

FB context. For illustrative purposes, based also on the findings of the preceding works,

Fig. 3 qualitatively illustrates such a drawback for distributed inelasticity frame elements,

rendering clear that the inelasticity concentration is dependent on the particular choices of

the mesh and/or the numerical integration scheme. Note how the numerical results

converge to a stable solution in the case of a hardening response, with the opposite

happening in the case of softening behavior.

The above numerical localization may be therefore regarded as fictitious because it

happens as a consequence of the FE assumptions. However, the term appears to be

appropriate because in both situations the result is that damage (real damage in the first

case, computed damage in the second) is localized into a limited zone. Logically, it would

be extremely useful to relate both phenomena—the physical one and the analytical—

which would require a sort of interface between the findings of fracture mechanics and

the FE formulation context. Unfortunately, such relation, though very appealing, is also

extremely difficult to find due to (1) the mentioned limitations of the former in com-

pletely understanding the physical phenomena (or, at least, in efficiently describing them)

and (2) the difficulties of the latter in including such findings.

Displacement-based FE that show softening behavior, such as, for instance, RC

columns modeled under a high axial load ratio and strong lateral loading, undergo

localization because the curvatures tend to concentrate into the mesh element subjected

to the highest bending moment. Considering the example of a cantilever under a constant

high axial load and a lateral (monotonic/cyclic) displacement history imposed to the top

FIGURE 2 Localization and size dependence in uniaxial and flexural compression

(based on Borges et al. [2004]).

42 A. Calabrese, J. P. Almeida, and R. Pinho

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free extremity, curvatures are expected to concentrate in the bottom (base) element. This

holds regardless of the mesh discretization that is adopted. That is, independently of the

number of subdivisions, the strains always localize in the base element. Additionally,

such features make the response nonobjective, because the computed curvatures will have

to increase at such base element when the mesh is refined, in order to maintain the same

value of the total displacement. Considering that the mesh can always be more refined (at

least, from the theoretical point of view), the curvatures can also always increase without

converging to any particular value. It will be seen later that, in fact, the localized response

occurs not only in the most strained element, as commonly accepted, but in particular at

the most strained controlling section within such element. At the global level the post-

peak response will also be nonobjective; i.e., different lateral force values are attained for

different meshes and the same imposed lateral displacement. The higher the number of

the mesh elements, the steeper the post-peak response.

Force-based elements are afflicted by an equivalent numerical shortcoming; i.e.,

strains concentrate at the integration point (IP) that first attains the peak bending moment.

If the above-mentioned cantilever is modeled with an FB element, the deformation will

localize at the base IP (near the fixed boundary). Recalling that for the FB approach—and

even for an inelastic response—no meshing is required, a one-to-one correspondence (one

FE per structural member) will always be adopted throughout the current work. The

nonobjective response in FB elements is thus characterized by an increase of curvatures at

the base IP as the number of IPs per element also increases. Such local behavior is

reflected in the global response of the member.

From the previous description it should be evident that both FB and DB elements,

when used to model softening behavior, require regularization procedures in order to

produce consistent and physically significant results. It is also apparent that this is

particularly important in performance-based earthquake engineering (PBEE), where

deformations and displacements of structural members have to be accurately predicted.

Δ

Δ

Δ

χ

χ

FIGURE 3 Nonlinear static analysis of a simple cantilever—objective and nonobjective

response corresponding to hardening and softening sectional behaviors.

Numerical Issues in Distributed Inelasticity 43

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4. Case Study

In order to illustrate the influence of the FE formulation, mesh discretization, and

numerical quadrature rule on the computed output, an experimental test is used as a

benchmark. The chosen specimen comes from the work of Tanaka and Park [1990],

where it is referred to as column unit No. 7. This is a largely referred RC cantilever of

1.65 m, loaded with an axial load ratio of 30% and cyclically tested in the horizontal

direction by an imposed displacement at the free node. The same test has also been

modeled by Coleman and Spacone [2001a] first and Scott and Fenves [2006] later. The

details of the specimen are reported in Table 1 and in Table 2. The OpenSees software

[Opensees, 2009] is used to perform all the analyses of this study.

4.1. Numerical Moment-Curvature Analyses

The transversal section has been discretized in 250 layers. For the concrete, the Kent and

Park [1971] model has been used. It has to be noted that, although more complex

hysteretic rules are available, the analyses herein presented are monotonic. Moreover,

this model can be easily modified in order to simulate a constant fracture energy, as

proposed by Coleman and Spacone [2001a]. Regarding the concrete parameters, the cover

concrete has been modeled with f 0c ¼ 32:1MPa, "0 ¼ 0:0024mm=mm, and

"20 ¼ 0:0248mm=mm. The confinement effect on the core concrete has been evaluated

with the Mander model [Mander et al., 1988], resulting in a confinement factor of

kc ¼ 1:215 with which the values f 0cc ¼ 39MPa—compression strength of the confined

concrete—and "c max ¼ 0:0052mm=mm—strain corresponding to the maximum concrete

stress—are obtained [these values are the same as those used by Coleman and Spacone,

2001a, and Scott and Fenves, 2006]. For what concerns the reinforcement steel, a bilinear

stress–strain relationship is used, with elastic modulus E = 200.000MPa, yield stress

fy ¼ 510MPa, and 1% strain-hardening ratio.

Figures 4 and 5 show the sectional response for two different axial load ratios: 30%

and 3% (corresponding respectively to an axial force of 2900kN and 290kN). The former

value is used in the experiment described above, whereas the latter will be applied in the

following sections to highlight the differences caused by a sectional softening behavior in

contrast to a hardening pattern.

From the comparison between the two curves it is immediate to detect the effect of

the axial load in both increasing the carried moment and reducing the available

sectional curvature ductility. In order to estimate the strain limits for moment-curvature

analysis, the proposals and formulas indicated by Priestley et al. [2007] have been

adopted. For the damage control compression strain, the confined concrete compression

strain limit is taken to occur when fracture of the transverse reinforcement confining the

core happens. For the present case this condition results in a value of "c;dc ¼ 0:0289. A

steel tension strain limit of "s;dc ¼ 0:6"su ¼ 0:06 is also considered. These two limits are

indicated in Fig. 4 and Fig. 5 below, showing that with low axial load ratio the steel limit is

reached first, while failure due to crushing of the concrete core controls for higher axial

load ratios.

5. Results Obtained With the FB Formulation

The response computed with an FB approach is presented for both hardening and soft-

ening conditions. The different behaviors of the element in the two cases are therefore

addressed. Different regularization procedures are then analyzed and compared.

44 A. Calabrese, J. P. Almeida, and R. Pinho

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5.1. Objective Response for FB Elements—Hardening Behavior

Very frequently, RC elements show essentially a hardening behavior. This leads to

an objective response, in the sense that the computed output will always converge

to a unique value. In order to show this characteristic, several analyses have been

performed under hardening conditions. Two hundred fifty layers are again used to

discretize the section. P-� effects were not considered in the analyses, in order to

provide a fair comparison between the formulations and between the theory and the

experimental results. The structural member has been modeled with a single FB element

(recall the one-to-one correspondence discussed in section 2.1). The Gauss-Lobatto quad-

rature rule was used to numerically integrate the FB elements. Figure 6 depicts the

nonlinear static curves comparison between different numbers of IPs.

It is possible to observe that all the lines get close to each other as the number of IPs

is increased, meaning that the response is objective at the global level. In fact, all the

curves show the same trend independently of the number of controlling sections. From Fig. 7

the convergence at the local level can also be assessed, as well as the exact satisfaction

0

200

400

600

800

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35Curvature [m–1]

Mom

ent [

kN*m

]

Steel damage control strain limitConcrete damage control strain limit

FIGURE 5 Moment-curvature diagram for 3% axial load ratio with indication of

damage control strain limits.

0

200

400

600

800

1000

0.00 0.10 0.20 0.30 0.40 0.50Curvature [m–1]

Mom

ent [

kN*m

]

Concrete damage control strain limit

Steel damage control strain limit

FIGURE 4 Moment-curvature diagram for 30% axial load ratio with indication of

damage control strain limits.

46 A. Calabrese, J. P. Almeida, and R. Pinho

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of equilibrium provided by the FB element (note that the moment profile should be a

straight line).

5.2. Nonobjective Response for FB Elements—Softening Behavior

A single element of 1.65 m is again used, and only the number of Gauss-Lobatto IPs is

changed for the various analyses. Under this increased value of the axial load ratio, the

global response is highly nonobjective, meaning that the results do not tend to a unique

value as the number of IPs is increased. The term localization is appropriate because all

the inelastic deformations concentrate at the base controlling section as soon as the

sectional response enters in the softening branch (i.e., negative tangent stiffness). As a

matter of fact, the computed curves are strongly affected by the number of IPs and do not

follow a single pattern after the peak global response is overtaken. This is depicted by

Fig. 8, the features of which clearly contrast with those of Fig. 6.

0

0.2

0.4

0.6

0.8

1

0Curvature [m–1]

4 Integration points

6 Integration points

8 Integration points

0

0.2

0.4

0.6

0.8

1

0400600800

Moment [kN*m]

Nor

mal

ized

col

umn

leng

th [

-]

200 0.150.05 0.1

FIGURE 7 Moment and curvature profiles at 2.5% drift (hardening response).

0

100

200

300

400

500

0.00 0.01 0.02 0.03 0.04 0.05Horizontal displacement [m]

Hor

izon

tal l

oad

[kN

]

3 Integration points

4 Integration points

5 Integration points

6 Integration points

FIGURE 6 Objective global response at 2.5% drift for FB formulation (hardening

response).

Numerical Issues in Distributed Inelasticity 47

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It is noted that such curves qualitatively match the ones obtained by Coleman and

Spacone [2001a] and reproduced later by Scott and Fenves [2006]. It is also important to

realize that this numerical output has no physical meaning, being simply an FE formula-

tion output. This is even more evident by looking at the sectional responses of Fig. 9,

where the localized curvatures attain unrealistic values for higher number of IPs.

In order to provide an immediate picture of the phenomenon, the moment and curvature

profiles up to eight IPs for a pre-peak and a post-peak condition are reported respectively in

Fig. 10 and Fig. 11. It is apparent that the post-peak behavior is characterized by a dramatic

localization of deformation and that the output generated by the FE formulation at this stage

has no physical meaning, both in terms of curvatures and moments. In fact, as can be detected

comparing the left plots of Figs. 10 and 11, the moment profile along the length of the

column is no longer captured after the peak response is attained. This is another consequence

of the loss of objectivity in the force prediction, already depicted in Fig. 8 and clearly

noticeable in Fig. 12. The element forces, and consequently the moment diagrams, depend on

the number of IPs once the softening branch is reached (Fig. 12), because element compat-

ibility has to be verified. This yields, in turn, the diverging moment profiles of Fig. 11.

5.3. Regularization Techniques for FB Elements

For the case of softening sectional responses, the need is evident for a regularization procedure

leading to consistent results; i.e., simultaneously objective and of physical value. Coleman

0 0.1 0.2 0.3 0.4 0.5 0.60

200

400

600

800

1000

Curvature [m–1]

Mom

ent

[kN

*m]

4 Integration points5 Integration points6 Integration points4 IPs end of the series5IPs end of the series6IPs end of the series

0 0.1 0.2 0.3 0.4 0.5 0.60

200

400

600

800

1000

Curvature [m–1]

Mom

ent

[kN

*m]

(i) (ii)

FIGURE 9 Cantilever base moment-curvature response for (1) 1.2% and (2) 3% drift.

0

100

200

300

400

500

600

0.00 0.01 0.02 0.03 0.04Horizontal displacement [m]

Hor

izon

tal l

oad

[kN

]

4 Integration points

5 Integration points

6 Integration points

FIGURE 8 Nonobjective global response for FB formulation.

48 A. Calabrese, J. P. Almeida, and R. Pinho

Downloaded By: [Eucentre] At: 23:59 27 August 2010

and Spacone [2001a] proposed a technique where the concrete uniaxial relationship is

modified, in order to maintain a constant fracture energy Gf. Assuming a value of Gf = 180

N/mm for the present case study, the inelastic curves reported in Fig. 12 can be obtained.

The procedure then requires post-processing the curvatures in order to obtain the

regularized sectional response, which is done geometrically assuming a plastic hinge

length, herein referred to as Lp. This length can be estimated, for instance, by using the

formula proposed by Paulay and Priestley [1992]:

Lp ¼ 0:08Lþ 0:022fydb ðkN;mmÞ ð1Þ

where fy and db are the yield strength and diameter of the longitudinal reinforcement. It

is highlighted that, although the previous quantity is a conventional parameter, it is

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8Curvature [m–1]

4 Integration points

6 Integration points

8 Integration points

0

0.2

0.4

0.6

0.8

1

0200400600800

Moment [kN*m]

Nor

mal

ized

col

umn

leng

th [

-]

FIGURE 11 Moment and curvature profiles at 2.5% drift (post-peak response).

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025Curvature [m–1]

0

0.2

0.4

0.6

0.8

1

02505007501000Moment [kN*m]

Nor

mal

ized

col

umn

leng

th [

-]

4 Integration points

6 Integration points

8 Integration points

FIGURE 10 Moment and curvature profiles at 0.6% drift (pre-peak response).

Numerical Issues in Distributed Inelasticity 49

Downloaded By: [Eucentre] At: 23:59 27 August 2010

taken in the present study as the actual length over which the real plastic curvatures

extend, also known as length of plastification. Scott and Fenves [2006] proposed a novel

plastic hinge integration method to restore objectivity for softening response. This proce-

dure is already available in OpenSees (BeamWithHinges element) and was used in the

present work (Figs. 13 and 14) for further comparison with experimental data.

If a specific regularization technique is not available, it is worthy to note that a fairly

common procedure in what concerns FB elements is to set the number of IPs in such a

way that the plastic hinge length is approximately equal to the length associated to the

base controlling section. By doing so, the physical aspect of localization could be

reflected in the numerical model. This possibility has the disadvantage that an error has

to be in most cases accepted for short elements, because the distance associated with the

base IP generally does not exactly coincide with the plastic hinge length. Moreover, in

case of long elements, additional nodes can be added to the frame model to adjust

the plastic hinge length with the base IP length but, obviously, the advantage of the FB

one-to-one correspondence is lost. Very short elements, such as the cantilever example of

this study, cannot in general be regularized in this way.

0

100

200

300

400

500

600

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Horizontal displacement [m]

Hor

izon

tal l

oad

[kN

]

Lp = 0.08L + 0.022fy·dbLp = 0.5·hExperimental (1st cycle)Experimental (2nd cycle)

FIGURE 13 Regularized global response up to 5% drift using the BeamWithHinges

element.

0

100

200

300

400

500

600

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Horizontal displacement [m]

4 IPs - Regularized5 IPs - Regularized6 IPs - Regularized4 Integration points5 Integration points6 Integration points

Hor

izon

tal l

oad

[kN

]

FIGURE 12 Regularized global response using the constant fracture energy approach.

50 A. Calabrese, J. P. Almeida, and R. Pinho

Downloaded By: [Eucentre] At: 23:59 27 August 2010

Finally, notice that not only the number of controlling sections can be changed but

the numerical integration rule could be defined to set a correspondence between the

plastic hinge length and the outermost IP length, in a similar fashion to that proposed by

Scott and Fenves [2006].

6. Results Obtained With the DB Formulation

Analogously to the previous section, the case study is here analyzed by making use of a

DB approach. The same analyses are performed and the consequences of this formulation

will be highlighted.

6.1. Objective Response for DB Elements—Hardening Behavior

As an initial test, the cantilever is modeled with a number of elements ranging from two

to ten. Each element is integrated with two Gauss-Legendre points. All elements have the

same dimensions within each discretization (i.e., 1.65/2, 1.65/3, 1.65/4 m and so on), with

no meshing refinements at the base, because the purpose of the analyses is only to

illustrate the effect of the FE formulation.

Figure 15 shows that the number of elements does not significantly change the global

results in the case of a positive sectional second stiffness, provided that at least four

elements are used. In fact, all the curves have the same trend and, more importantly, converge

to a unique response as the number of elements per member increase—objective response.

Notice that, in the elastic range of behaviors, a single-element mesh would also provide the

same results. The differences can be detected in the nonlinear range of behaviors. Here,

increasing the number of elements generally produces a lower, more realistic, estimation of

the lateral load. This has to be so, because increasing the number of elements yields a more

flexible model. It is important to realize, however, that though the global response can be

satisfactorily predicted with any of the discretizations with more than three DB elements,

none of them would provide suitable estimates of the local quantities. That could only be

achieved with a much finer mesh, as indicated by Figs. 16 and 17.

0

200

400

600

800

1000

0.000 0.025 0.050 0.075 0.100 0.125 0.150 0.175 0.200Curvature [m–1]

Mom

ent [

kN*m

]

Lp = 0.08L + 0.022fy·db

Lp = 0.5·h

Concrete damage control strain limit

Experimental data

Theoretical monotonic prediction by Tanaka and Park [1990]

FIGURE 14 Regularized local response up to 5% drift using the BeamWithHinges

element.

Numerical Issues in Distributed Inelasticity 51

Downloaded By: [Eucentre] At: 23:59 27 August 2010

Figure 16 also shows an interesting feature that does not occur for the FB formulation

(see Fig. 9): the sectional moment-curvature curves show some scatter, although they

refer to (approximately) the same cross section under the same constant value of imposed

axial load on the cantilever. Intuitively, a perfect match between all the curves could be

expected (regardless of the actual values of the moment-curvature position being depen-

dent on the mesh discretization). Another observation comes from the moment profiles

depicted in Fig. 17, which are not coincident. The reason for both features lies in the fact

that DB elements satisfy equilibrium only in an average (integral) sense. This aspect will

be extensively addressed later in this article.

6.1.1. Efficiency of the Numerical Integration Scheme. In addition to the influence of the

mesh discretization, the effect of the numerical quadrature rule is investigated. Figure 18

shows the consequence of the integration scheme in the computed global response.

A mesh with three DB elements of length Lcol/3 is used. The use of two Gauss-

Legendre points, or three Gauss-Lobatto points, allows the exact integration of a polynomial

0

100

200

300

400

500

600

700

0.00 0.03 0.06 0.09 0.12 0.15Curvature [m–1]

Mom

ent

[kN

*m]

2 DB Elements

3 DB Elements

5 DB Elements

10 DB Elements

200 DB Elements

FIGURE 16 Local response at 2.5% drift for DB formulation (hardening response) in

the bottom IP.

0

50

100

150

200

250

300

350

400

450

0.00 0.01 0.02 0.03 0.04 0.05Horizontal displacement [m]

Hor

izon

tal l

oad

[kN

]2 DB Elements3 DB Elements4 DB Elements5 DB Elements6 DB Elements7 DB Elements8 DB Elements9 DB Elements10 DB Elements

FIGURE 15 Objective global response at 2.5% drift for DB formulation (hardening

response).

52 A. Calabrese, J. P. Almeida, and R. Pinho

Downloaded By: [Eucentre] At: 23:59 27 August 2010

of the third degree. The other two schemes (four Gauss-Legendre IPs and five Gauss-Lobatto

IPs) provide the exact result for a polynomial of the seventh degree. As is evident from the

plot, no effect of the integration scheme can be detected at this level by looking at the element

response, even when a relatively crude mesh of three DB elements is used. To check whether

the influence of the number of IPs was being ‘‘hidden’’ by the meshing of the cantilever, it was

decided to model the member with a single DB element; see Fig. 19.

Obviously, such modeling is not intended to suitably represent the inelastic behavior

of the element (which is apparent from the comparison with the ten DB-elements mesh

results). Although the curves corresponding to different number of IPs can now be

0

100

200

300

400

500

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09Horizontal displacement [m]

10 DB Elements (2 Gauss-Legendre IPs)

2 Gauss-Legendre integration points

4 Gauss-Legendre integration points3 Gauss-Lobatto integration points

5 Gauss-Lobatto integration points

Hor

izon

tal l

oad

[kN

]

FIGURE 18 Three DB elements (L/3 each)—effect of the integration scheme in the

computed global response (note that the lines corresponding to two through five IPs are

virtually coincident).

0

0.2

0.4

0.6

0.8

1

0 0.05 0.1 0.15

Curvature [m–1]

2 DB Elements

3 DB Elements

5 DB Elements

10 DB Elements

200 DB Elements

0

0.2

0.4

0.6

0.8

1

0200400600800Moment [kN*m]

Nor

mal

ized

col

umn

leng

th [

-]

FIGURE 17 DB formulation: moment and curvature profiles at 2.5% drift (hardening

response).

Numerical Issues in Distributed Inelasticity 53

Downloaded By: [Eucentre] At: 23:59 27 August 2010

distinguished, the fact that no major differences are recognizable indicates that two

Gauss-Legendre points are sufficient, in any case, to integrate with enough accuracy

the DB-element quantities. Figure 20 shows the results for a two DB-element mesh and

different number of IPs. It is no longer possible to distinguish the different series, which

further supports the previous observation. However, notice that the mesh chosen is still

too crude to provide satisfactory results.

Based on the above considerations, it can be stated that, for sectional hardening

behavior and DB elements (1) global results are objective; (2) apparently there is no

justification to use more than two Gauss-Legendre IPs per element; (3) a mesh of at least

four elements per member is required to provide sufficiently accurate results at the global

level. Therefore, throughout the following analyses a constant number of Gauss-Legendre

IPs per element equal to two will be used—this is already the case of the models used to

get an insight into the local behavior of the element, depicted in Fig. 17.

0

100

200

300

400

500

600

700

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09Horizontal displacement [m]

10 DB Elements (2 Gauss-Legendre IPs)2 Gauss-Legendre integration points 3 Gauss-Legendre integration points 4 Gauss-Legendre integration points 5 Gauss-Legendre integration points

Hor

izon

tal l

oad

[kN

]

FIGURE 19 One DB element—effect of the different number of IPs.

0

100

200

300

400

500

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09Horizontal displacement [m]

10 DB Elements (2 Gauss-Legendre IPs)2 Gauss-Legendre integration points3 Gauss-Legendre integration points4 Gauss-Legendre integration points5 Gauss-Legendre integration points

Hor

izon

tal l

oad

[kN

]

FIGURE 20 Two DB elements (L/2 each)—effect of the different number of IPs (note

that the lines corresponding to two through five IPs are virtually coincident).

54 A. Calabrese, J. P. Almeida, and R. Pinho

Downloaded By: [Eucentre] At: 23:59 27 August 2010

6.2. Nonobjective Response for DB Elements—Softening Behavior

In section 5.2, the localization phenomenon for FB elements was presented. However,

because the concentration of deformations depends on the softening response of the

sectional constitutive relation, it is expected that a similar occurrence can also be found

with the more classical stiffness formulation. Setting the axial load to the real value used

in the experiment, localization effects do show up at the global level (Fig. 21). Moreover,

a finer discretization yields a steeper descending branch, which is clearly symptomatic of

the concentration of deformations. Again, within the elastic range of behaviors of the

element, there is no influence of the mesh on the calculated response.

Similarly to what already happened for the hardening condition, in this case there is a

marked divergence in the local quantities’ predictions, as can be detected from the

moment-curvature diagrams at the bottom Gauss section in Fig. 22.

It can be noted that the latter moment-curvature curves are not coincident. From a

conceptual point of view, only the point associated with the 2.5% drift along the moment-

curvature curve should change, not the curve itself (recall that the section and the

imposed value of the cantilever axial load are the same in all cases). The reason behind

such discrepancy is again the fact that the DB formulation does not strictly verify

equilibrium. This characteristic implies that the damage control strain limit turns into a

range of values; such a damage control bound for concrete is indicated with a grey shade

in Fig. 22. Finally, by looking at the profiles obtained for the softening response (Fig. 23),

it is possible to appreciate that there is significant variability in both moment and

curvature distributions.

6.3. Regularization Techniques for DB Formulations

Although DB elements are still more widely employed than FB ones, a limited number of

regularization procedures are found in the literature. It is usual to regularize them by

assuming that the extremity elements’ lengths are equal to a multiple of the plastic hinge

length. It is commonly believed that the use of displacement interpolation functions force

localization within a single element instead of one IP. Such an underlying principle

justifies the regularization criterion of defining the length of the most strained element

Hor

izon

tal l

oad

[kN

]

0

100

200

300

400

500

600

700

0.00 0.01 0.02 0.03 0.04Horizontal displacement [m]

4 DB Elements

6 DB Elements

8 DB Elements

10 DB Elements

FIGURE 21 Nonobjective global response at 2.5% drift for DB formulation (softening

response).

Numerical Issues in Distributed Inelasticity 55

Downloaded By: [Eucentre] At: 23:59 27 August 2010

as equal to the plastic hinge length, as mentioned in the introduction. Applying this

criterion to the case study cantilever yields the ‘‘regularized’’ global response of Fig. 24,

which is compared to the experimental results.

The findings from the present work (see also section 7.2, Fig. 34), however, seem to

indicate that, at least for the current DB model, localization effectively occurs within the

bottom IP, similarly to FB elements. This behavior was also encountered in one of the first

0

0.2

0.4

0.6

0.8

1

0 0.1 0.2 0.3Curvature [m–1]

4 DB elements

6 DB elements

8 DB Elements

10 DB Elements

0

0.2

0.4

0.6

0.8

1

0200400600800Moment [kN*m]

Nor

mal

ized

col

umn

leng

th [

-]

FIGURE 23 DB formulation: moment and curvature profiles at 2.5% drift (softening

response).

0

200

400

600

800

1000

1200

0.00 0.05 0.10 0.15 0.20 0.25Curvature [m–1]

Mom

ent [

kN*m

]

2 DB Elements

4 DB Elements

6 DB Elements

8 DB Elements

10 DB ElementsCon

cret

e da

mag

e co

ntro

l str

ain

lim

it

FIGURE 22 Nonobjective local response at 2.5% drift for DB formulation (softening

response) in the bottom IP.

56 A. Calabrese, J. P. Almeida, and R. Pinho

Downloaded By: [Eucentre] At: 23:59 27 August 2010

studies on the topic [Zeris and Mahin, 1988]. Considering the case of two Gauss points per

element, this means that the length of the extremity elements should therefore be defined as

equal to twice Lp and not only Lp as generally accepted. This effectively means that the

distance corresponding to each IP, where inelasticity will concentrate, is the same as

the plastic hinge length. In other words, a physically consistent regularization technique

for the DB formulation requires an intuitively paradoxical definition of the most strained

mesh elements’ lengths as being distinct from the plastic hinge length (or, to be more

precise, the length of plastification). In this way, the correspondence between the analytical

and the physical deformations is kept, because the inelastic response will be integrated

along the appropriate distance. In fact, if such extremity mesh elements’ lengths are exactly

equal to the plastic hinge length, the curvature will be overestimated, as will shortly be

seen. For the cantilever under analysis, this theoretically derived principle does not manage

to provide a very satisfactory estimate of the global behavior; see Fig. 25. However, this

latter result can be considered as artificially misleading because the ratio between the base

0

100

200

300

400

500

600

700

800

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09Horizontal displacement [m]

Hor

izon

tal l

oad

[kN

]

2 DB Elements, Length Bottom El. = 2 x Lp2 DB Elements, Length Bottom El. = LpExperimental (1st cycle)Experimental (2nd cycle)

FIGURE 25 Softening case; comparison between different bottom DB element lengths

(2 Lp and Lp). Global regularized response up to 5% drift.

0

100

200

300

400

500

600

700

0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09Horizontal displacement [m]

Hor

izon

tal l

oad

[kN

]

2 DB Elements, Length Bottom El. = Lp3 DB Elements, Length Bottom El. = Lp4 DB Elements, Length Bottom El. = Lp5 DB Elements, Length Bottom El. = LpExperimental (1st cycle)Experimental (2nd cycle)

FIGURE 24 Softening case. DB regularized response up to 5% drift with bottom DB

element length = Lp.

Numerical Issues in Distributed Inelasticity 57

Downloaded By: [Eucentre] At: 23:59 27 August 2010

element’s length (=0.71m) and the total column height (=1.65m) is considerably larger than

the expected equivalent ratio for common-sized RC members.

The confirmation of such criterion validity emerges through the comparison of the

local response, which is incomparably closer, in terms of 5% drift point, to the experi-

mental results, as indicated by the ending curvatures depicted in Fig. 26.

Therefore, when modeling softening behavior with DB elements, the possibility of

defining the length of the most strained elements (generally, the extremity ones) as 2·Lp

instead of simply Lp should be critically assessed. Although the previous considerations

assume two Gauss-Legendre points per element, it is noted that the use of a different number

of IPs would imply an adjusted length associated to the most strained elements, according to

the same rationale. It is believed that this alternative regularization procedure for DB

elements should be considered by engineers and researchers that use software packages

where this type of formulation is implemented. However, the proposed approach, though

adequate for softening conditions, could be unsuitable for modeling elements with hardening

sectional behavior. In this case, in fact, a greater accuracy is obtained by reducing the size of

the element subjected to the largest excitation, as will be analyzed in detail in section 7.1. It is

again observed that the novel regularization technique presented in this work was based on

the plastic hinge length in view of the fact that this value has often been used for the same

purpose in similar studies. However, the influence on the accuracy of the results regarding

the use of such conventional quantity, versus the employment of a physical characteristic

length like the so-called length of plastification, should be thoroughly investigated. Other

general considerations on the use of a plastic hinge length as a regularization parameter can

be found in Calabrese [2008].

7. Comparison Between FB and DB Elements

The first example aims to underline the fundamental differences between the two

formulations, recalling the discussion presented in section 2.1. The response of a simple

inelastic analysis considering sectional hardening behavior is depicted in Fig. 27, for

which a two-element mesh with two Gauss-Legendre IPs per element has been used.

As expected, though the overall trend is the same for the two formulations, the DB

model is artificially stronger than the FB one, and also stiffer, as soon as the element goes

0

250

500

750

1000

0.00 0.05 0.10 0.15 0.20 0.25Curvature [m–1]

Mom

ent [

kN*m

]

2 DB Elements, Length Bottom El. = 2 x Lp

2 DB Elements, Length Bottom El. = Lp

Experimental data

Theoretical monotonic prediction by Tanaka and Park [1990]

Lbottom elem = 2LP

Lbottom elem = LP

FIGURE 26 Softening case; comparison between different bottom DB element lengths

(2 Lp and Lp). Moment-curvature regularized response up to 5% drift.

58 A. Calabrese, J. P. Almeida, and R. Pinho

Downloaded By: [Eucentre] At: 23:59 27 August 2010

into the inelastic range. Similar considerations on the renowned effects caused by the two

formulations were also presented by Coleman and Spacone [2001b].

For what concerns the softening case, it can be observed in Fig. 28 that the

differences in the post-peak branches of the DB elements are reduced with respect to

those of their FB counterparts. The discrepancies in the DB results begin to be more

detectable increasing greatly the number of subdivisions. Regarding the numerical inte-

gration procedure, it has to be recalled that in these examples the Gauss-Legendre

quadrature rule is used, which does not include any IP at the base of the cantilever. On

the contrary, when illustrating the flexibility formulation, the Gauss-Lobatto rule sets an

IP at the base of the column, capturing in this way the highest moments and the highest

deformations. For this reason, the nonobjectivity obtained with the DB formulation is

‘‘regularized’’ by the element dimension and by the absence of a controlling point at the

more strained section.

Hor

izon

tal l

oad

[kN

]

0

100

200

300

400

500

600

700

0.00 0.01 0.02 0.03 0.04Horizontal displacement [m]

4 DB Elements6 DB Elements8 DB Elements10 DB ElementsFB Element - 4 IP sFB Element - 5 IP sFB Element - 6 IP s

FIGURE 28 Nonobjective global responses at 2.5% drift for DB and FB formulation

(softening response).

0

100

200

300

400

500

0 0.02 0.04 0.06 0.08 0.1Horizontal displacement [m]

Hor

izon

tal l

oad

[kN

]

DB formulation

FB formulation

FIGURE 27 Comparison of the DB and FB inelastic curves of the test column (hardening

response).

Numerical Issues in Distributed Inelasticity 59

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7.1. Local Convergence for FB and DB Formulations

Figures 16 and 17 indicate that the curvature estimation provided by the DB formulation,

even under hardening conditions, is affected by a sort of localization, because an extremely

large number of subdivisions is required in order to converge to a unique value. It is therefore

useful to assess the curvature at the bottom IP as a function of the number of DB elements

used to discretize the cantilever. Figure 29 refers to the curvature at the bottom controlling

section resulting from a nonlinear static analysis up to 2.5% drift under hardening conditions.

The outcomes from FB and DB models are plotted in the same graph. The FB results are

obtained by using a single element and a varying number of Gauss-Lobatto IPs. Therefore,

the two horizontal axes in the figure refer respectively to the number of DB elements (bottom

axis) and to the number of Gauss-Lobatto IPs for the FB element (top axis). Observe also that

the axis reporting the number of elements is in logarithmic scale, whereas the other is linear.

It is noted that the curvatures obtained with the FB formulation are at the bottom

Gauss-Lobatto IP; i.e., they are computed exactly at the base section of the cantilever. On

the other hand, the curvatures from the DB analyses are all associated with a different

abscissa, because they correspond to the bottom Gauss-Legendre section in the bottom

element of the mesh (this difference becomes negligible with increasing number of

elements; therefore, general comments and comparisons are valid).

Several features become immediately noticeable from the two plots of Fig. 29. As

expected, the curvature values obtained with a single FB element are generally larger than

the ones predicted by the DB models. This result confirms that the FB element does not

restrain the development of inelastic deformations, as happens with the DB element due

to the imposed linear curvature distribution. Furthermore, both formulations eventually

converge to a unique curvature value (around 0.14 m�1), meaning that from the theore-

tical viewpoint the response is objective for both cases.

Anyway, the very large number of elements required by the DB mesh to approach the

final asymptotic value allows one to consider the issue of ‘‘practical localization’’; i.e., a

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1001# Elements, DB

Cur

vatu

re [

m–1

]

2 3 4 5 6 7 8 9 10# IPs, FB

DB formulation

FB formulation

FIGURE 29 Hardening case. Curvature at the bottom IP at 2.5% drift for different mesh

refinements (DB) and numbers of IPs (FB).

60 A. Calabrese, J. P. Almeida, and R. Pinho

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nonobjective prediction that will be encountered if a ‘‘common engineering mesh’’ is

used (note that the finest mesh—200 elements—corresponds to 8-mm-length elements).

The current study shows that this phenomenon can be much more pronounced than

indicated in the previous work of Neuenhofer and Filippou [1997].

Summarizing, the FB formulation is definitely more efficient than the DB one, also in

what concerns the prediction of curvatures, because only a few IPs are sufficient to attain a

converged response (in this case, six IPs provide an acceptable result at the local level).

The accurate sensitivity of the FB formulation is also responsible for producing

marked localized results in the case of softening systems. Figure 30 refers to the nonlinear

static analysis up to 2.5% drift on the cantilever loaded to 30% of its axial capacity (note

that in this plot both horizontal axes are in linear scale).

As previously discussed, both FB and DB formulations provide global and local

nonobjective responses in the case of softening sectional behavior. In both cases the

computed curvatures increase more than linearly with the number of IPs per element (FB

case) or the number of elements per member (DB case). Figure 30 shows that the localized

effects are more pronounced for the FB formulation, at least considering the ‘‘engineering

range of variation’’ commonly used in structural earthquake engineering models.

7.2. Other Considerations on the DB Element (Hardening and Softening Behaviors)

In the previous sections it has been noted that the sectional response computed with the DB

formulation did not follow the sectional moment-curvature relationship represented in Figs.

4 and 5. Additionally, it varied as a function of the mesh subdivision. Apparently this may

seem incorrect, because the section does not change and the axial load imposed on the

cantilever is also the same. In other words, one could think that the number of DB elements

could affect the stage reached along each computed moment-curvature curve but never the

curve itself. Moreover, even with a hardening response and from a practical point of view, a

localized curvature prediction has been individuated along with an objective global

response. These two aspects, which are due to the same cause, are now analyzed.

0.05

0.25

0.45

0.65

0.85

1.05

1.25

1 2 3 4 5 6 7 8 9 10# Elements, DB

Cur

vatu

re [

m–1

]

2 3 4 5 6 7 8 9 10

# IPs, FB

DB formulation

FB formulation

FIGURE 30 Softening case. Curvature at the bottom IP at 2.5% drift for different mesh

refinements (DB) and numbers of IPs (FB).

Numerical Issues in Distributed Inelasticity 61

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A major difference between FB and DB elements is that, for the former, equilibrium

is strictly satisfied (in fact, it is enforced), whereas for the latter equilibrium is verified

only in an average sense. This means that in the FB formulation the sectional forces (axial

forces and moments) are in exact equilibrium with the external values. On the contrary, in

the DB formulation the equilibrium is not verified point-wise, because it is the weighted

integral of the internal forces that has to equilibrate the element forces. Using FE

terminology, equilibrium is satisfied in a strong form for FB elements but only in a

weak form for DB ones.

The simplest way to assess this fundamental difference is by observing the sectional

axial forces of a beam-column element. For FB elements, these forces are always equal to

the applied axial load on the element, regardless of the material constitutive relationship.

On the contrary, in the DB case the applied axial load is equal to the weighted integral of

the section axial forces. Therefore, as long as the material has an elastic behavior (and loads

are applied at the extremities of the element), the sectional axial forces are exactly the same

as the axial load acting on the column. Instead, for nonlinear material behavior, the section

axial forces will in general not be equal to the applied axial load, as illustrated in Fig. 31.

This figure refers to inelastic analyses up to 5% drift with an axial load ratio of 3%.

A two-element mesh (0.5 m + 1.15 m) with two Gauss-Legendre IPs per element is used

and the corresponding graph shows the results of the bottom IP of the element at the

column’s base. It is apparent that for the FB formulation the sectional axial force at the

base controlling section is strictly in equilibrium with the external axial load throughout

the analysis; i.e., also in the nonlinear range. Oppositely, as soon as the elastic range of

behaviors is surpassed, the axial force at the bottom controlling section of the base DB

element starts increasing.

Figure 32 illustrates the same feature by depicting addionally the evolution of the

axial force at the top section of the same element. The similar but ‘‘mirrored’’ pattern that

can be found is readily justified because the weighted sum of the top and bottom axial

forces has to equal the imposed external element load. The symmetry of Fig. 33 is a

reminder that the integration weight at both sections is 0.5.

Finally, the above considerations explain the differences in the moment-curvature

paths highlighted when commenting on the results of Figs. 16 and 22. The most important

consequence of such inconsistency is that it is conceptually incorrect to evaluate the

moment-curvature sectional response from a fiber-section DB element (because it does

depend on the element characteristics), even for a hardening behavior. On the other hand,

DB formulation

250

300

350

400

450

0.00 0.02 0.04 0.06 0.08Top node horizontal displacement [m]

Forc

e [k

N]

Axial load on the element

Axial force at the bottom controlling section

FB formulation

250

300

350

400

450

0.00 0.02 0.04 0.06 0.08Top node horizontal displacement [m]

Forc

e [k

N]

Axial load on the element

Axial force at the bottom controlling section

FIGURE 31 Sectional axial force versus element axial load for both formulations up to

5% drift in the bottom IP.

62 A. Calabrese, J. P. Almeida, and R. Pinho

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0

200

400

600

800

0.00 0.02 0.04 0.06

Curvature [m–1]

Sect

iona

l mom

ent [

kN*m

]

0

100

200

300

400

500

600

0.00 0.02 0.04 0.06

Curvature [m–1]

Sect

iona

l axi

al f

orce

[kN

]

Bottom DB element, upper IPBottom DB element, bottom IPBottom DB element, weighted sum

FIGURE 32 Sectional axial force and moment recorded at the base DB element up to

2.5% drift (hardening response).

150

200

250

300

350

400

450

0.00 0.02 0.04 0.06 0.08Top node horizontal displacement [m]

Forc

e [k

N]

Axial force at the upper controlling sectionAxial force at the bottom controlling sectionWeighted sum

FIGURE 33 Axial forces at the two controlling sections of the bottom DB element.

Numerical Issues in Distributed Inelasticity 63

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the moment-curvature sectional response extracted from an FB element is the correct one;

i.e., the same that would be predicted by a moment-curvature analysis on the isolated section.

Another interesting observation comes from the local response of two integration

sections belonging to the same DB element. Figures 32 and 34 (respectively hardening

and softening conditions) refer to a nonlinear static analysis up to 2.5% drift. In both

cases the cantilever is modeled with two DB elements, each of them integrated with two

Gauss-Legendre sections.

It is immediate to detect how, for both values of the axial load, the moment-curvature

curves within each element are significantly dissimilar. This is physically untenable,

because the section has the same geometrical and mechanical characteristics at the two

IPs, and the axial force applied to the cantilever is constant. This consideration holds for

hardening and softening conditions although it is significantly more evident for the latter

case. Figure 34 indicates that the upper section starts to unload at a curvature of about

0.022 m�1, in correspondence with an increase in the sectional axial force. It is noted that

the element with a hardening response does not experience such numerical unloading.

Recall that for the DB element, all the local patterns (for hardening and softening

conditions) are severely conditioned by the FE formulation and thus have an obviously

reduced physical meaning. For the softening case, all but the bottom integration sections

undergo unloading, which is an important observation for the construction of regulariza-

tion procedures for the DB formulation, recalled next.

0

200

400

600

800

1000

1200

0.00 0.02 0.04 0.06 0.08 0.10Curvature [m–1]

Sect

iona

l mom

ent [

kN*m

]

Lower integration section

Upper integration section

Moment peak response

Sectional axial force starts to increase

2000

2500

3000

0.00 0.02 0.04 0.06 0.08 0.10Curvature [m–1]

Sect

iona

l axi

al f

orce

[kN

]

Sectional axial force at the upper integration section

Moment peak response

Sectional axial force starts to increase

FIGURE 34 Sectional axial force and moment recorded at the base DB element up to

2.5% drift (softening response).

64 A. Calabrese, J. P. Almeida, and R. Pinho

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8. Summary of Numerical Issues

The observations derived from the analyses performed on the case study are herein

summarized, in order to provide a clear compendium that can be considered by engineers

and researchers for an accurate modeling of RC structures.

It is of paramount importance to know the type of element formulation (FB or DB)

available in the software package being used. The DB approach predicts unrealistically

higher element stiffness and strength compared to the exact FB formulation, which is a

well-known feature among FE researchers [e.g., Coleman and Spacone, 2001b;

Neuenhofer and Filippou, 1997].

As theoretically expected, DB elements only manage to verify equilibrium in an

average sense. This issue is responsible for two other drawbacks of this formulation.

Firstly, since in general a strict equilibrium is not verified, the combination of forces

acting in each section can be considerably far off the exact equilibrium with real loading

conditions, thus causing an unreliable or, at least, a ‘‘slowed-down evolution’’ of local

quantities. Secondly, if the sectional constitutive model is obtained via a fiber model—

which takes into account the interaction between axial force and bending moment—this

effect is even more dramatic for it gives rise to a sectional behavior that, ultimately, is

dependent on the number of elements per frame.

In the presence of a hardening sectional response, a Gauss-Lobatto integration scheme

should be used for FB elements. Although a lower bound of four IPs per element is required

in order to provide a reliable result at the global level (and a satisfactory response at the local

level), a choice of a larger number of, for instance, six IPs can also be justifiable in order to

obtain a completely stabilized prediction of the local response. It can thus be inferred that the

objective response predicted by FB elements rapidly converges as the number of IPs is

increased, as was also found by Neuenhofer and Filippou [1997].

Still referring to hardening conditions, the local response of a DB element is also

objective, yet the convergence (as a function of the number of elements per member) is

much slower than the corresponding one for the FB. In fact, it is so slow that, for the usual

range of number of elements per member commonly used in structural analysis, strongly

localized results are produced: the value of the curvature for four DB elements is less than

50% of the objective value. This is naturally of the greatest concern, from the viewpoint of

engineering practice, because it implies that local measures obtained from a DB model

(such as curvatures or strains) can be heavily underestimated. The present case study

showed in a straightforward way that this delayed convergence can be much more

significant than what Neuenhofer and Filippou [1997] previously pointed out. It is under-

lined that such a conclusion does not apply to the global output, which is expected to be

accurate for a mesh with few DB elements.

The numerical analyses performed in this study confirmed the lack of conver-

gence in case of softening sectional response, in accordance to seminal literature

references [Coleman and Spacone, 2001a; Zeris and Mahin, 1988]. In fact, when the

peak of the sectional behavior is surpassed, the global and local responses of either

DB or FB formulations become nonobjective. As the number of IPs per element (FB

formulation) or the number of elements per member (DB approach) augments (1) the

global post-peak branch becomes steeper and (2) the curvatures concentrate at the

bottom IP.

In order to overcome this issue, several regularization procedures for FB elements

have been proposed in the literature [e.g., Coleman and Spacone, 2001a; Lee and

Filippou, 2009; Scott and Fenves, 2006; Scott and Hamutcuoglu, 2008]. If the use of

one of these procedures for FB elements is not available, an alternative is still possible by

Numerical Issues in Distributed Inelasticity 65

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specifying the length associated with the extreme IP such that it is close to the plastic

hinge length. This can be done by selecting an appropriate number of IPs or even by

choosing a particular numerical integration rule. Such an approach is not very practical

and, for standard dimensions of beams, columns, and plastic hinge lengths, it generally

requires some degree of approximation. It may also require meshing in more than one

element per member, therefore weakening one of the main advantages of the FB

formulation.

A commonly used regularization technique for DB meshes is accomplished by

specifying the size of the frame extremities’ elements as equal to the appropriate

equivalent plastic hinge length. However, this approach overestimates the actual inelas-

tic deformations. This is due to the fact that, unlike what is generally assumed,

localization in the DB mesh occurred in the most strained IP and not in the whole

element. Based on this consideration, a novel approach is suggested to regularize the

response of DB elements. This requires the length of the farthest mesh element to be

larger than the plastic hinge one, depending on the particular integration scheme

adopted.

9. Conclusions

The initial main objective of the present work was to collect disperse information

into a single and consistent manuscript tackling the influence of the main para-

meters affecting the performance of distributed inelasticity models. From the

application viewpoint concerning structural engineering practice, it is the authors’

expectation that the conclusions and comments carried out throughout the study can

be taken into account by engineers and researchers for an accurate modeling of RC

structures.

Several numerical analyses, based on an experimental test widely referred to in

recent literature, have been performed. The consequences of different modeling assump-

tions, such as element formulation, mesh discretization, number of IPs, and numerical

integration scheme, were investigated and systematized for hardening and softening

sectional behaviors.

From the analyses performed, a larger-than-expected delay in convergence for the

objective local response of DB elements should be highlighted. Additionally, for the

usual engineering range of number of IPs per element (FB case) and number of elements

per member (DB case), the localized response is more apparent in the FB formulation

than in the DB one. In other words, although in both cases the response is nonobjective

and untrustworthy, the FB local response shows a considerably higher scatter of base

curvatures in comparison to the DB output.

Despite its eminently practical nature, this work also questions commonly estab-

lished concepts. In particular, it was shown that, in the case of DB formulation and

softening conditions, the localized response does not actually concentrate at the element

level but at the IP level as in FB elements. This consideration led to the proposal of a new

physically meaningful regularization technique where the size of the frame extremities’

elements is set to be larger than the plastic hinge length.

The validity of such rationale was confirmed by comparison with the experimen-

tally derived local response. The small dimensions of the cantilever under study caused

a misleading disagreement in terms of the global response, but such deviation is

certainly reduced for normal-sized RC members. Thus, if two Gauss-Legendre points

per element are used, the ‘‘regularized’’ length of the extremity elements should be

twice the value of the plastic hinge length. By doing so, the physical meaning of the

66 A. Calabrese, J. P. Almeida, and R. Pinho

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local response under softening behavior is naturally reflected in the numerical integra-

tion of the sectional quantities.

Acknowledgments

The first author gratefully acknowledges the Italian Air Force for the research leave to

join the Rose School–IUSS. The second author is grateful for the financial support

provided by the Fundacao para a Ciencia e Tecnologia through the funding program

Programa Operacional Ciencia e Inovacao 2010.

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