Elsevier Editorial System(tm) for Journal of Constructional Steel Research Manuscript Draft Manuscript Number: Title: LOCAL DUCTILITY OF STEEL BEAMS UNDER FAR-FIELD EARTHQUAKES Article Type: Research Paper Keywords: Local ductility, rotation capacity, local plastic mechanism, far-field earthquake, number of cycles, ultra low cycle fatigue, cyclic ductility classes. Corresponding Author: mr., dr., prof., eng. victor gioncu, Ph.D Corresponding Author's Institution: Politehnica University Timisoara First Author: victor gioncu, Ph.D Order of Authors: victor gioncu, Ph.D; ANTHIMOS ANASTASIADIS, Dr. Eng; MARIUS MOSOARCA, Ass. Prof. Manuscript Region of Origin: ROMANIA Abstract: In the framework of a modern multi-level earthquake design the association of the engineering seismology with the structural design is emphasized; the first one providing the loading characteristics while the second one the behavior of the structural elements. The paper is focused on the two aforementioned topics. The first part briefly presents the main earthquake types defined as far-source and near-source seismic actions and further is concentrated on the far-field earthquakes which are characterized by a predominately repetitive cyclic loading effect on structural systems. Also the issue of duration as well as the type and the number of cycles in case of far-field earthquakes is discussed. The second part deals with the investigation of the cyclic available ductility, in the frame of the new concept of Ultra Low Cycle Fatigue, based on the concept of the plastic collapse mechanism, previously developed for monotonic loading [3],[4], in combination with the introduction of the initial cumulative deformation concept. The study is oriented to evaluate the main parameters affecting the cyclic available rotation capacity of steel I-beams, concluding that the loading type defined by the increasing or constant amplitude as well as the number of cycles producing strength and ductility degradation and cross section conformation are of primary importance while, in general, the steel quality and material variability have a secondary detrimental effect regarding the steel members framing structures excited by far-source ground motions.

Cover letter

The author for correspondence:

Dr. Eng. Anastasiadis Anthimos

Contact address:

Romania, Timisoara, 3 Toplita Street, Postal Code: 300012; Telephone/fax: 0040256226277;

Mobile : 0040740470642; E-mail: anastasiadisa@hol.gr

Previous papers:

(1) Gioncu, V., Petcu, D.: Available rotation capacity of wide-flange beams and beam-

columns. Part1, Theoretical approaches. Journal of Constructional Steel Research,

1997, 43, No. 1-3, pp. 161-218.

(2) Gioncu, V., Petcu, D.: Experimental and numerical tests. Part 2. Journal of

Constructional Steel Research, 1997, 43, No. 1-3, pp. 219-244.

(3) Gioncu, V., Mosoarca, M., Anastasiadis, A.: Prediction of available rotation capacity

and ductility of wide-flange beams: Part 1. DuctRot-M computer program. Journal

of Constructional Steel Research, 2012, 69, pp. 8-19.

(4) Anastasiadis, A., Gioncu, V., Mosoarca, M.: Prediction of available rotation capacity

and ductility of wide-flange beams: Part 2. Applications. Journal of Constructional

Steel Research, 2012, 68, pp. 176-191.

The previous papers published in 1997 and 2012 set the base of the plastic collapse mechanism in

order to predict the available ductility of I-shaped steel beams, under monotonic loading

conditions, as well as the development of a computer program, namely the DuctRot-M (Ductility

Rotation-Member), facilitating the calculation of the member rotation capacity. Using the

aforementioned concept a series of parametrical studies was carried out revealing some practical

conclusions regarding the conformation of welded as well as hot-rolled I sections.

The present paper goes further, being a continuity of the previous one published in 1997 and

2012, focused on the effect of cyclic action as defined by the far-field seismic action. According

to the multi-level earthquake design combines the aspects of engineering seismology with the

inelastic behavior following the path of source-ground motion characterization, local soil

conditions, type of induced seismic action, local member structural response and briefly discus

the above mentioned topics. Moreover, introduce and implement the concept of the initial

cumulative deformation, as the accumulated initial imperfection, to the plastic collapse

mechanism in order to study the effect of cyclic action on the available rotation capacity of hot-

rolled I steel beams. Using the model that takes into account the cyclic action, as introduced in

the DucRot-M software, a parametrical analysis was performed analyzing the main parameters

affecting the local ductility of steel beams.

Cover Letter

Highlights

> Far-field earthquake characteristics introducing a predominately cyclic action are

briefly discussed. > The concept of the initial cumulative deformation, by the analogy

with the initial elastic imperfection, for the prediction of the local ductility under

cyclic loads is presented. > The plastic collapse mechanism theory in association with

the initial cumulative deformation was used for the study of I-shaped beams. >A

parametrical analysis reveals the main factors affecting the cyclic ductility, namely

the loading amplitude, the number of cycles, the cross-section conformation. > Cyclic

ductility is obtained by using a correction factor, as the effective number of cycles

producing erosion, applied on the monotonic ductility.

*Research HighlightsClick here to download Research Highlights: Research Highlight.doc

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LOCAL DUCTILITY OF STEEL BEAMS

UNDER FAR-FIELD EARTHQUAKES

Anthimos Anastasiadisa, Marius Mosoarca

b, Victor Gioncu

b

a ASA Structural Consultants, Thessaloniki, Greece

b “POLITEHNICA” University Timisoara, Timisoara, Romania

ABSTRACT

In the framework of a modern multi-level earthquake design the association of the engineering

seismology with the structural design is emphasized; the first one providing the loading

characteristics while the second one the behavior of the structural elements. The paper is focused on

the two aforementioned topics. The first part briefly presents the main earthquake types defined as far-

source and near-source seismic actions and further is concentrated on the far-field earthquakes which

are characterized by a predominately repetitive cyclic loading effect on structural systems. Also the

issue of duration as well as the type and the number of cycles in case of far-field earthquakes is

discussed. The second part deals with the investigation of the cyclic available ductility, in the frame of

the new concept of Ultra Low Cycle Fatigue, based on the concept of the plastic collapse mechanism,

previously developed for monotonic loading [3],[4], in combination with the introduction of the initial

cumulative deformation concept. The study is oriented to evaluate the main parameters affecting the

cyclic available rotation capacity of steel I-beams, concluding that the loading type defined by the

increasing or constant amplitude as well as the number of cycles producing strength and ductility

degradation and cross section conformation are of primary importance while, in general, the steel

quality and material variability have a secondary detrimental effect regarding the steel members

framing structures excited by far-source ground motions.

Keywords: Local ductility, rotation capacity, local plastic mechanism, far-field earthquake, number of

cycles, ultra low cycle fatigue, cyclic ductility classes.

1. INTRODUCTION

Seismic-resistant structures are usually designed relying on their ability to sustain high plastic

deformations. The design philosophy considers that the earthquake input energy is dissipated through

the hysteretic behavior of a member; plastic hinges are formed in predetermined positions due to a

number of cycles of seismic loading. This concept is based on the condition that the plastic hinge must

show a stable hysteretic behavior with a sufficient rotational ductility to allow for dissipating this input

energy. Before the 1960s the ductility notion was used only to characterize the material behavior.

After the Housner’s studies regarding earthquake problems and Baker’s research works on plastic

design, this concept has been extended to a structural level. According to this design philosophy, the

structure may be designed for lower forces than those it has to resist, taking into account the inelastic

reserves of the structural system. Therefore, the evaluation of the available ductility is of primary

importance. The use of the monotonic ductility for seismic actions has provided to be a valuable

concept in many earthquakes and it corresponds to the methodology included in the modern codes. For

instance, the Eurocode 8 [1] specifies the use of the ductility classes prescribed in Eurocode 3 [2],

mainly determined for static design. This approach is based on the observation that for many cases the

load-deformation skeleton curves (constructed using the cyclic curve) correspond very well with the

monotonic curves. The ductility of steel members for monotonic loads has been studied by Gioncu et

al [3] and Anastasiadis et al [4].

Corresponding author. E-mail address: anastasiadisa@hol.gr

*ManuscriptClick here to view linked References

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Unfortunately, the 1977 Vrancea and the 1985 Mexico City earthquakes, both far-field earthquakes

with relevant soft soil conditions, as well as the 1994 Northridge and 1995 Kobe earthquakes, both

characterized as near-field events, provoked unexpected damage that have seriously compromised the

validity of the code provisions, with respect to required and available ductility. In this direction they

unveil a significant gap in the knowledge of seismic behavior of steel structures in the extreme

conditions of seismic loading. Therefore, after these series of devastating earthquakes, it has been

recognized by society that both seismic hazard and risk have to be reassessed.

This target is possible only if the impressive progress in Seismology will be transferred by

Engineering Seismology into Earthquake Engineering. The basic concepts of today’s seismic codes

were born almost 70 years ago, when the knowledge about the seismic actions and structural response

were rather poor. Nowadays, the earthquake-resistant design is grown within the new multi-

disciplinary fields of Engineering Seismology and Earthquake Engineering, wherein many exciting

developments are produced and are predicted to the near future. Therefore, it is very clear that any

progress is impossible without considering the new amount of knowledge recently cumulated in

Seismology due to the fact that the same structural system behaves in a different manner as a function

of earthquake type. Lessons learned from past 40 years of real earthquake excitations reveal that in

seismic design the following should be considered: (i) The ground motions characteristics in function

of source-site distance (far-field and near-field earthquakes); (ii) The source types (interplate,

intraplate and intraslab) with very different rupture characteristics. There are so big differences

between the ground motions generated by these sources at different epicenter-site distance that

ignorance of these aspects can be considered as a shortcoming in seismic design. In this context, the

Engineering Seismology is now paying more attention to establish the differences in the main

characteristics of the sources. At the same time, the task of Earthquake Engineering is to take more

care about the structural response for different ground motions.

The basic conclusion of the recent research works for seismic design is that the structures must be

designed in function of the position from source (far or near-field) and the main ground motion

characteristics (e.g. cyclic or pulse seismic actions, number of cycles or pulses, duration, etc).

From this point of view the earthquakes can be classified as ordinarily, for ground motion

characteristics regularly included in code provisions, and exceptionally, when these characteristics are

not explicitly considered in codes. The Mexico City, Northridge and Kobe earthquakes belong to the

last case. Current design practice is based only on the earthquake magnitude and the corresponding

spectrum, however does not assure a proper and safe seismic design for all earthquake types [5], [6].

In the past, due to the reduced number of records during severe earthquakes, mainly obtained far

from recording stations, the codified design methodologies were developed on the consideration that

the ground motions are characterized only by large number of reversal cycles in accelerograms and

pure site soil condition influence. During the very last years, due to the development of a large

network of instrumentation all over the world, there are a large amount of records of ground motions

for different epicentral distances as well as different local site conditions. The analysis of this new

information has emphasized the diversity of ground typologies. For instance, these new information

offers the possibility to consider, for design purposes, the great differences in the ground motions

between far-source and near-source seismic regions. In spite of this situation, the effects of near-field

versus far-field ground motions are not well understood. Moreover, until now, structural design codes

do not recognize the principal behavioral difference among the aforementioned earthquake types.

Each event is basically unique due to the influences of many different factors. The earthquake can

be considered as the result of the behavior of a nonlinear system which is extremely sensible to the

very small changing of initial conditions, approached by the theory of Chaos and the science of

Seismology [5]. Therefore, the determination of the actual characteristics is a very difficult task.

As a consequence, the seismic actions are determined with great uncertainty. But the structure must

be endowed by design with the ability to develop and maintain its bearing capacity, even when the

considered seismic action exceeds the design limits. A measure of this ability is the ductility, which is

the structural performance to sustain these exceeding by large deformations in plastic range without

significant loss of resistance. The basic ductility design criterion, where any earthquake-resistant

structure must satisfy, is the following [3]:

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Required ductility < Available ductility

The required ductility is the effect of earthquake on the structure, determining the maximum values of

seismic demand for design purposes. The available ductility is the structure’s ability to resist that

effect without failure.

The general aspects of required and available ductility, in function of earthquake types are

presented in [7] and [8]. The main conclusion of these studies is that exist very large differences in

post-elastic response in case for considering the effects of far-field and near-field earthquake loading

as well as the source type. The main characteristics of near-field versus far-field earthquakes in

determining the required ductility are presented in many papers [5],[9],[10],[11],[12]. However,

referring to the available member ductility, the studies, relatively, are very few.

For far-field earthquakes, characterized by a large number of reversal cycles and accumulation of

plastic deformations, the available ductility is presented in [13] and [14]. For near-field earthquakes,

for which the seismic loads are characterized by velocity pulses, reduced number of reversal cycles

and influence of strain-rate, some results are presented in [13].

Generally, the prediction of the available ductility should take into account the fundamental

differences between ground motions. In this context, the simplest methodology is to evaluate the

ductility for monotonic loads and accordingly to correct the determined values considering the specific

characteristics of each earthquake type:

Seismic ductility = Correction factor x Monotonic ductility

The consideration of the monotonic ductility as basic value and not the one resulting from the low

cyclic fatigue actually frames in the current results obtained using the new fatigue category Ultra Low

Cyclic Fatigue (see Paragraph 3.5), which is devoted to the case of very reduced number of cycles

(e.g. until 20-30 cycles). The complexity of the analysis regarding the local ductility capacity directly

related to the nature of earthquakes forced to divide the study in two parts; however the present paper

is devoted to the influence of far-field earthquakes, while a companion paper [15], which is under

preparation, will be referred to the near-field earthquakes, examining the impact of velocity pulses and

strain rate on steel beams.

In this direction, the present paper interconnects some aspects of engineering seismology, trying to

get some evidence regarding the loading conditions, with the one of steel earthquake-resistant

structures. Moreover, focused on the prediction of the available member ductility, and based on the

concept of the monotonic ductility [3], [4], evaluates the seismic available rotational capacity of steel

members considering the case of far-field earthquakes. However, the aforementioned seismic

excitations are characterized by a repetitive cyclic action, due to a leading role of the site soil

conditions, which produce an accumulation of plastic deformation. Accordingly, the study emphasized

the influence of the number of reversal cycles on the available ductility of steel members

2 FACTORS DETERMINING THE EARTHQUAKE TYPES

2.1 Source types, source depth, source-site distance and site conditions

Each earthquake is unique, being the result of the effect of many factors influencing the ground

motions, in this way generating very complex phenomena. At first glance examining the recorded

ground motions a chaotic movement can be observed without showing any rule. Nevertheless, a

seismologist looking to an accelerogram, can detect the anatomy of an earthquake. Contrary to the first

impression, the ground motions are the result of the overlapping of the effects of a limited number of

basic factors, having different importance and putting their imprint on the recorded movements. Due

to the fact that an interaction between these factors exists, this different but related parts of ground

motions can be, in some extent, identified. The main factors influencing the ground motions are

source type, source–site distance and site conditions.

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In function of the source types and depth, there are the following main earthquake types (Fig.1) [5]:

- Crustal earthquakes, produced along the tectonic convergent plate boundaries (inter-plate

earthquakes) or in the interior crust of a tectonic plate (intra-plate earthquakes). The depth of

these earthquakes do not exceed 40-50 km. The most important type is the inter-plate

earthquake which is produced by subduction (subduction of an oceanic plate under a

continental one), collision (converging of two continental plates) or strike-slip (two tectonic

plates slide and grind against each other). These earthquakes are dominated by mechanical

processes (shear forces between the two tectonic plates).

- Sub-crustal earthquakes, occur in slab (subducted oceanic plate), the source depth being

situated in range of 50-300 km. They are situated under the crust, where the solid rock begins

to be transformed into molted-lava, due to the high temperature and pressures. Therefore the

deep earthquakes are dominated by thermal phenomena, while the intermediate ones, until 150

km, by a combination between mechanical and thermal ones.

The depth of source is a very important factor, because the crustal earthquake effects are limited

only for short distances around the epicenter on a small area, while the subcrustal sources may

produce great events in region far from the epicenter, affecting a large area (Fig. 2).

The source-site distance plays a leading role in the design of structures and a classification is

absolutely necessary. In function of this distance the following classification may be considered (Fig.

3) [13]:

- Near-field site, with a concentrated zone including the area around the epicenter, generally with

a radius equal to the source depth.

- Far-field site, with a distance about 4-5 times the source depth. In this type the intermediate

site, defined as a case between near and far-field site, is also included.

Site conditions can be responsible for micro-seismic variation, which can be more important than

the source-site distance (Fig.4). There are the following site conditions [5]:

- Local horizontal layered deposit, which can be identified by characterizing the multi-layers

with different mechanical properties and thickness, corresponding to some soil categories.

- Topographic surface irregularities, where amplification of seismic response is produced in top

of hills.

- Alluvial basins, when the basin produced an important amplification of the source

accelerations.

- Liquefaction, produced by loss of strength in saturated, cohesionless soils due to the build-up

of pore water pressure during dynamic loading.

For the classification of earthquake type, the distance from the site to the source is the most

adequate measure.

2.2 Far-field earthquakes from crustal sources

Crustal earthquakes (subduction or strike-slip types) occur in shallow sources (Fig. 5). The

earthquake characteristics rely on the propagation-path of the body P and S waves, surface L and R

one as well as local soil conditions.

Firstly, the propagation path effect depends on the percentage of the path travel through soft

sediments. Deviations from a uniform horizontal layered crust model, along the path of the waves

propagating from the epicenter to the site, are occurred. These deviations are produced by a collection

of sedimentary basins with alluviums, separated by irregular basement rock, forming mountains as

well as geological and topographical irregularities. Therefore, two opposed phenomena occur, the first

one is the attenuation of ground motions compared to the distance from the epicenter and the second

one is the amplification of these motions due to presence of the sedimentary deposits. Due to these

effects, the recorded acceleration, velocities and displacements are very different in near and far-field

sites (Fig. 6). Generally, the near-field records are related to velocity pulse with reduced number of

cycles and very short earthquake duration (produced by P and S waves), while for far-field, the

records are characterized by many acceleration cycles and long duration of earthquakes (produced by

L and R waves). For intermediate sites the earthquakes must have the characteristics of both far and

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near-field earthquakes. Even though it is very well known that the body waves, P-S, are characterized

by high frequencies, while the surfaces waves, L-R, by low frequencies, and further by the fact that the

higher frequencies attenuate more rapidly than the lower one as the distance becomes longer.

Consequently, in intermediate sites the earthquake characteristics are dominated by the far-field

earthquake characteristics. It should be mentioned that for this earthquake type the site soil conditions

play a leading role, in many cases being more important than the earthquake type or traveled path.

Secondly, the soil profile is composed by multi-layers with different mechanical properties and

thickness. The alternate of layers is a very important factor which has been long recognized; the nature

of layers change the amplitude and frequency with a direct effect on the degree of structural damage.

2.3 Earthquakes from sub-crustal sources

Sub-crustal earthquakes are produced by deep sources (Fig. 7). In this case the earthquake

characteristics are given as a function of both L and R surface waves, P and S body waves and finally

the reflected and refracted waves. Contrary to crustal earthquakes, in this mentioned situation the

earthquake characteristics depend on the succession of vertical layers and not on horizontal travel

path. In addition, the effects of near-field source disappear, the earthquake characteristics being

practically the same over an extended surface. Therefore, for this type it is difficult to discuss about

the near or far-field earthquakes. In any case, due to the great depth of source, the main characteristics

correspond very well with far-field earthquakes, as duration, number of cycles, influence of soil site

conditions, etc.

3 DUCTILITY UNDER CYCLE LOADING AFFECTED BY FAR FIELD EARTHQUAKES

Usually, for engineering purposes earthquake ground motions are characterized by their

acceleration amplitudes. A more complete description of the ground shaking also requires the

introduction of an indicator as the ground shaking duration and the number of cycles of motion [16].

Only in this way it is possible to determine the structural damage in which the effect of the available

ductility is included [17].

3.1 Shaking Duration

Ground motion duration is an important parameter for the seismic design of a structure situated in

far-field sites. However, this parameter is not yet taken into account in the seismic codes due to the

fact that the evaluation of the effective duration of an earthquake and the influence of this duration on

the structural behavior are a very complex task. There are several hundred papers in literature related

to the influence of duration of strong ground motions on structural damage, however with different

conclusions. Some of them suggest that the duration should be incorporated into the specification of

seismic design, but there are also opinions that supports the policy of current anti-seismic codes to

neglect the duration effect [18]. Evidently, from the engineering point of view, it is very clear that in

case of two earthquakes having the same magnitude but different durations, the damage are more

important for the one having greater duration.

Recently, the increased network of recording stations offers the possibility to overcome this

problem. The duration is defined by braked duration (the time interval between the first and the last

time when the acceleration exceeds the level of 0.05g), uniform duration (the length of time for which

the ground motion acceleration exceeds a fixed threshold value), and significant duration (the length

of time interval between the two time points when the Arias intensity exceeds two separate fixed

threshold values) [19]. The seismic design must include the effects of duration by the number of

cycles, because long duration has been recognized as a potential source of structural damage. It is

well-known that the structure’s strength is not related to the ground motion duration, to the contrary

the earthquake duration, as a result of the cumulative plastic deformations, have a significant effect on

the available ductility.

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In addition, the ground motion duration depends on the duration of the rupture process at the source,

prolongation of the duration due to propagation path effects, and prolongation effects caused by the

local soil layers at the recording site [20], [21]. Figure 8 presents the increasing of duration in function

of earthquake magnitude, and source-to-site distance [13]. It can be observed that the duration

increasing is proportional to the epicentral distance and further that the duration is greater for large

magnitudes.

Generally, the duration in far-field sites can be framed in the interval of 30 to 50 sec. During this

interval the structures are subjected to a large number of cycle loadings, but only a reduced number of

them have important effects damaging the structure.

3.2 Effective number of cycles of earthquake ground motions

The number of cycles of ground motions is widely used in the geotechnical earthquake engineering

(liquefaction and landslide problems), although not for the structural analysis. In spite of that, all

specialists recognize that the number of cycles is a robust indicator associated with the structural

destructiveness of an earthquake, however there is no universally accepted approach to determine this

indicator of damaging shakings [22].

There are two main methodologies, the first one defined by a direct counting of the dominating

cycle using ground motion’s time history, and the second one by a cycle counting using the structural

response time history. Figure 9 shows very clearly that the number of cycles is related not only to the

ground motions but also to the structural response. The number of cycles is larger for rigid structures

than the flexible one. Hence, the second methodology is more reliable. Dividing the total hysteretic

energy by the equivalent energy that would be absorbed by a section of structure (Fig. 10a) [23], it can

be obtained the effective number of cycles of an earthquake ground motion. Figure 10b, which is an

application of the above mentioned methodology, shows the equivalent number of cycles for the case

of an inter-plate action like the Northridge earthquake. The main observation from this figure is that

the number of cycles for far-field sites is about 10-12 cycles.

Unfortunately, due to the complexity of problem, there are not systematic research works which

estimate the number of cycles in function of the main influencing factors such as, source type,

earthquake magnitude, structural rigidity (rigid structures, T< 1 sec, flexible structures, T >1 sec). For

preliminary design of structures situated in far-field regions and taking into account the main

characteristics of different earthquake types [5],[13], it is possible to consider the following number of

effective cycles:

Earthquake type Magnitude Rigid structures Flexible structures

- Intra-plate earthquakes M < 6.0 max 5 cycles -

- Inter-plate earthquakes:

subduction M < 9.0 max 10 cycles max 5 cycles

strike-slip M < 7.0 max 12 cycles max 6 cycles

- Intra-slab earthquakes M < 8.0 max 15 cycles max 8 cycles

3.3 Typology of cycle loading

The effect of cyclic actions on ductility, in case of structures situated in far-field areas, is developed

by the reduction of rotation capacity due to the accumulation of plastic deformations.

The seismic actions are characterized by a first period with a slow increasing of accelerations. After

the culminating phase, a decreasing of ground motions occurs until the movement completely stops.

The effect of this process is the formation of a global mechanism, composed of sufficient number of

plastic hinges (Fig. 11a) If θp is the rotation for which plastic rotation occurs and θb the rotation for

which plastic buckling of compression flanges takes place, then three basic behavior types can be

distinguished (Fig. 11b, Fig. 12):

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- when structural sections work in the plastic range without buckling (at a rotation level between

θp and θb) , an accumulation of plastic rotations occurs without any degradation in moment

capacity of plastic hinge;

- when the flange buckling is produced at the maximum seismic action (at a rotation level of θb)

, the degradation of plastic hinge works with constant amplitude;

- when the flange buckling, θb, is produced before reaching the maximum seismic action, the

degradation of the plastic hinge works with the increasing of amplitude for each cycle.

Therefore, in order to study the influence of cyclic loading on the ductility reduction two types of

cyclic actions should be considered, namely under constant (Fig. 12a) or increased amplitudes (Fig.

12b). Generally, for far-field earthquakes, the type of cyclic action is strictly connected with the

effects of soil conditions. Therefore for the prediction of the ductility reduction, it is recommended to

be used the following rules:

- for normal soil conditions (soils Class A, B and C, corresponding to EC 8 rules), cyclic actions

under constant amplitude;

- for soft soil conditions (soils Class D, E, S1,,2 corresponding to EC8 rules), cyclic actions under

increasing amplitude.

Different loading protocols are proposed trying to simulate the earthquake excitation; for instance,

the ECCS Recommendations [24], for increasing actions, suggest a repetition of three times each cycle

(Fig. 12c). It is important to underline that ECCS procedure considers that the step of experimental

cyclic action is related to the plastic rotation θp . In any case the aforementioned issue remains an open

task under debate, because the actual behavior is very far from the rules of recommendation.

3.4 Main effects of cycle loading

Seismic design philosophy accept the approach that significant excursions into inelastic range will

occur during severe earthquakes. However current codified methodologies, which adopts an elasto-

plastic model, does not make a direct reference to the effects of plastic accumulation provided by

cycle loading.

There are to main effects according to the capacity-demand approach, the first one referring to the

increasing of required ductility, the second one to the degradation of the available ductiltity. Figure

13a plots the effect of accumulation of plastic rotation on the required rotation capacity while the

Figure 13b presents the erosion of available rotation capacity, due to the action of multiple cycles [13].

The erosion refers to the moment and rotation capacities; this paper deals only with ductility problems.

Therefore, the main effect of cycle loading, due to the accumulation of plastic deformations, which

can be observed in any moment-rotation curve, is a degradation of ductility with the respect to the one

determined under monotonic loading.

3.5 Low-cycle fatigue or accumulation of plastic deformations?

In material science, fatigue is the progressive and localized structural damage that occurs when a

material is subjected to cyclic loading.

There are two fatigue types, high cycle fatigue (HCF) and low cycle fatigue (LCF) (Fig. 14).

Traditionally fracture mechanics and in some special cases damage mechanics was used for the

investigation of high-cycle fatigue problems in order to predict fracture in elastic range as well as the

initiation of a crack. The high-cycle fatigue occurs for more than 104 cycles and the minimum values

for fatigue verification are obtained for 106 cycles, being characterized by Woehler curve The low-

cycle fatigue instead occurs close (or at) the yield limit in elasto-plastic range under maximum 104

cycles, characterized by Manson-Coffin low.

In the last decades the effects of earthquake cycle loadings was attempted to be studied in the frame

of low-cycle fatigue rules, using the Manson-Coffin low. But from the recorded experimental data it

is observed that this law does not fit well in the range of very low number of cycles, i.e. about less

than 100 cycles [25]. The experimental curves are concave smoothly toward horizontal axis,

corresponding to monotonic loading. Recently, a new category, namely, ultra low cycle fatigue

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

(ULCF), (named also extremely low cycle fatigue, ELCF) is proposed for maximum 20 cycles (Fig.

14) [25],[26], recognizing the importance of reduced number of cycles.

For civil engineering structures and especially for building structures, some members may undergo

stresses in plastic range when they are cyclically loaded; in many cases the damage being caused by

plastic buckling. Considering the maximum effective number of cycles (less than twenty), it is obvious

that the effects of cycles produced by seismic excitation belongs to the lower limit of ultra-low-cycle

fatigue. As a consequence, the damage curves for earthquake effects are similar with the one of

monotonic loads [25]. The low cycle fatigue is characterized by a slope, corresponding to the reducing

effects of cyclic loading. In exchange, the ultra low cycle fatigue shows a tangent curve to the

horizontal line [26], corresponding to monotonic loading. This observation strengthens the basic

concept and proves the methodology proposed in this paper to use the results obtained for monotonic

ductility, with the proper implementation of correction factors taking into account the specific

characteristics of earthquake type. In the same time, considering the reduced number of effective

cycles during earthquakes a question arises, namely, to frame these effects in the category of fatigue or

it is more valuable to consider these one only as a result of accumulation of plastic deformations.

Low-cycle fatigue is connected with crack initiation-propagation and mainly to brittle fracture of a

tension flange. However, in case of strengthening (e.g. using ribs, cover plates) or weakening (e.g.

reduced beam section) of the connection, where the plastic hinge is moved away from the column

face, the flange plastic buckling works as a fuse dissipating energy and avoiding brittle fractures [27],

[28]. This was demonstrated from experimental evidence [27]. In fact, the conceptual approach of this

topic is strictly related to the detailing of the potential region of the formation of plastic hinge.

Moreover, the concept of plastic accumulation looks more open, can be combined well with the

established buckling theories and finally overcomes the experimental constraints of the fatigue

approach.

4. CYCLIC ACTIONS ON STEEL I-SHAPED BEAMS

4.1 Review on experimental studies and theoretical approaches

For steel beams the most usual cross-section is the I-shaped section. Nevertheless, the effects of

cyclic actions are studied for these sections, but the results are valuable also for other sections.

A detailed discussion of all the available literature is beyond the scope of this paper; a compilation

of the existing results on steel beams, however, is of primary importance. Therefore, for the shake of

completeness, a concise review of the experimental and theoretical work done in a so complex field,

which still remains open, is provided. The examined experimental tests are presented in Table 1.

4.2 Experimental testing

The experimental tests conducted by Bertero and Popov [29] on rolled steel sections showed three

important parameters: (i) the importance of controlling deformations at each cycle; (ii) evidenced the

influence of number of cycles to fracture; (iii) the effect of flange local buckling which can cause a

rapid reduction in the number of cycles until fracture.

The influence of lateral buckling on steel rolled beams subjected to repeated and reversal loading

was examined by Takanashi [30] revealing a great reduction of the rotation capacity as the lateral

slenderness ratio increases (for the examined cases between 51.20-81.70) obtaining values of cyclic

rotation capacity of 1.0 to 2.50 respectively for low carbon steel. Furthermore, the lateral instability

effect differs between monotonic and cyclic loading, the ductility is reduced drastically in the second

loading type. Guruparan and Walpole [31] also performed tests on hot rolled and welded I-shaped

beams showing that for lateral slenderness of 37 the strength degradation is acceptable, for a value of

57 the deterioration was greater while for the larger value of 97 a twisting was observed and the

reduction was not acceptable.

Tests by Van et al [32], Mitani et al [33], Suzuki and Ono [34], indicated that the deterioration is

severe only when local flange buckling is accompanied with the web buckling or lateral-torsional

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

buckling. Web buckling or flange buckling produce strength degradation while the lateral torsional

buckling produce a loss of stiffness or deterioration of the load carrying capacity. In addition, in many

cases the moment capacity under repeated cyclic action is a little higher than the monotonic one, due

to strain-hardening effect, but always a drastic deterioration after local buckling is remarked when the

beam was subjected to cyclic actions. Suzuki and Ono [34], from their tests, recorded cyclic rotation

capacities, as a function of the lateral slenderness, between 3.0-9.80 for the ratios of about 65, 55, 45

respectively. Unfortunately, their paper does not provide detailed information about the geometrical

and mechanical characteristics of the tested specimens. Lee and Lee [35] also noticed the strength

degradation after local buckling, showing that the deformation capacity of cyclically loaded beams is

reduced about 60% compared with the monotonic one.

In Europe at the Politecnico di Milano [36],[37],[59],[73],[74] an important experimental program

was performed investigating the flange local buckling of I-shaped steel welded and hot-rolled sections

under predominantly cyclic bending moments applying different increasing displacement histories as

well as the ECCS Recommendations [24]. Among the interesting results the following was remarked:

(i) the indication that once the flange has buckled, the maximum load cannot be reached in the

subsequent cycles; (ii) the amplification of local flange buckling give rise to a significant decrease of

loading carrying capacity; (iii) cracks appearing at the flange-web welded junction tends to extend up

to the flange tip. Another testing program, at the same university, [38],[59], [75] focusing on the local

ductility of European I-shaped sections (IPE, HEA, HEB) subjected to different cyclic loading

histories, as proposed by ECCS [24], was carried out revealing that IPE sections, having a comparable

flange slenderness ratio with that of HEB, have shown a reduced ductility due to the increased web

slenderness. Reanalysis of the aforementioned experimental programs by Castiglioni [39] evidenced

that a premature brittle failure may occur if the cycle amplitude is not large enough to produce flange

local buckling. Gioncu & Petcu [40] had already mentioned that local buckling works as a fuse

preventing early undesirable failure mechanisms.

Tests by Takanshai and Udagawa [41], Castiliogni et al [42] and Valente & Castiglioni [43] on steel

beems and composite members indicated that the presence of the slab reduces the cyclic inelastic

capacity, due to the fact that the upper flange local buckling is prevented. As a consequence in highly

stressed locations it is possible to appear early fractures as it was confirmed by real earthquake events

(e.g. Northridge, 1994, Kobe, 1995).

The influence of yielding strength on the cyclic ductility of steel beams was experimentally

examined, among others, by Takanashi [30] and Green et al [44, 45]. The tests showed that the

increasing of the yield strength or the yield ratio decreases the cyclic inelastic capacity of the member.

Jiao et al [46] revealed the important contribution of the Bauschinger effect on the plastic energy

dissipation capacity of steel beams, which is a basic factor of difference between monotonic and cyclic

loading.

Gioncu et al [47] performed a series of tests focusing on the study of the buckled shapes that

formed under monotonic and cyclic loads. They showed that in the second load case a superposition of

two plastic mechanisms occurs as a result of reversed bending action. Moreover, Mateescu & Gioncu

[48] examined the influence of loading types, as defined by an increased, decreased and pulse

displacement. The first two actions develop a plastic accumulation while the third one early fractures.

4.3 Theoretical approaches

For the examination of the influence of cyclic loading on steel members five principal approaches

were proposed.

The constitutive law method, which considers the formulation of constitutive laws capable of

reproducing the inelastic response of members in the presence of damage, plotted in terms of force-

displacement curves, Castiglioni [49], Castiglioni et al [50]. An alternative method was used by Vayas

[51] based on the material law and associated with an extension of the methodology of effective width,

already examined for monotonic loads, in cyclic loading in order to predict the strength and stiffness

degradation. Zambrano et al [52] propose a constitutive law with degrading strength and stiffness

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

characteristics taking into account the damage accumulation and the maximum excursion in the

inelastic range.

Numerical methods. The strip method, in which the cross-section of the member is divided in a

finite number of strips, each strip being characterized by area, distance from centroid, residual stress,

yield stress and ultimate strain [53]. To account for the member damage, due to local buckling,

fracture and gradual degradation, it is assumed that the strip area is reduced according to a certain law

depending on the type of structural damage. FEM analysis is also used in many research efforts [54],

[55], [56].

Low cycle fatigue approaches. Yamada et al [57] and Yamada [58] proposed low cyclic fatigue

fracture limits for all the element components in order to describe the structural ductility. Using the

classical rule of the Mason-Coffin, Ballio and Castiglioni [59] and Castiglioni et al [60], developed a

procedure that unifies the high and low cyclic fatigue for the design and damage assessment of steel

members. Recently, Lee and Stojadinovic [61], by making use of the plastic collapse mechanisms,

developed a method that considers the strength degradation induced by local buckling and low-cycle

fatigue fracture exploiting the principals of the above mentioned Manson-Coffin rule.

Energy based approaches, mainly expressed from Japanese structural community, in which the

collapse of a member is limited by an energy criterion. Akiyama propose a criterion in energy terms,

defining that a failure is possible to be occurred when the cumulated plastic ductility, given by the

skeleton curve, is grater than the monotonic one [62].

Plastic deformation accumulation approaches. Generally, there are two directions accounting for

plastic accumulation. The first one is developed based on the theory of damage models, defining the

failure as a limiting value of deterioration and further using the Miner’s assumption of linear damage

accumulation, properly adjusted, for the attainment of collapse condition [63],[64]. The second one is

based on the plastic collapse mechanism model, where flange and web local buckling are obtained

after the accumulation of residual displacements, provided by cyclic action, and as a result developing

a moment-rotation curve with softening branch describing the gradual degradation after each cycle

[65].

4.4 Concluding remarks extracted from experimental and theoretical results

(i) Experimental results concerning I-shaped sections have shown a very high sensibility to the

cyclic actions, for both strength and rotation capacities. The loading tests were governed by a

large number of cycles, exceeding the maximum number of cycles expected during an

earthquake.

(ii) There are not enough experimental data to be used for developing a comprehensive approach

for effects of cycle loading in the frame of the new concept of the ultra low cycle fatigue.

(iii) Experimental evidence has revealed that it is possible to make use of the monotonic ductility,

with proper reductions, in order to predict the cyclic ductility. Therefore, the experimental

and theoretical works previously examined provides the basic effects for the correction of the

monotonic ductility.

(iv) Commenting on available theoretical approaches, it could be noted that methodologies based

on low-cycle fatigue or damage models are generally computational cumbersome and are

associated with fatigue curves which should be developed for a series of sections and details.

FEM analysis, when properly calibrated, provides accurate results but it is not suitable for the

practical design due to big computing time as well as difficulties and uncertainties regarding

discretization. Energy based approaches is physically compatible with the seismic action,

however it is difficult to handle and in this way is also relatively impractical for design

purposes. Finally, the plastic collapse mechanism methodology seems to be simple and

accurate based on classical theories of plastic and buckling analysis and will be further

analyzed.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

5. EROSION OF MONOTONIC DUCTILITY DUE TO ACCUMULATION OF PLASTIC

DEFORMATIONS

5.1 Accumulation of plastic deformations

Throughout the cyclic loading a specific plastic mechanism is observed, characterized by the

superimposition of two plastic mechanisms under the opposite action of bending moments (Fig.

15,18,19,20). In order to implement the effect of the accumulation of plastic deformations in the hole

cyclic process, a conceptual similarity between a classical stability issue and one of the cyclic inelastic

action is proposed. During the formation of this plastic mechanism the compressed flanges buckle in

plastic range, where after cycle i - 1 remains a clear deformed local plastic shape (Fig. 15a). Moreover,

for the cycle i a new plastic deformation, in the form of local mechanism, superimposes over the

deformed shape corresponding to i-1 cycle. Consequently, as in case of stability problems, the

deformed shape of i–1 cycle can be considered as a geometrical imperfection (initial geometrical

deformation), over which will superimposed the plastic deformations, corresponding to i cycle (Fig.

15b).

Therefore, the effects for a stipulated number of effective cycles (see Section 3.2), can be

approached as a continuum accumulation of plastic deformations, resulting in a corresponding

reduction of strength and rotation capacity.

5.2 Example for a bended plate under cyclic loading

In order to demonstrate the aforementioned concept a bended plate which forms a plastic

mechanism composed by three yield lines is considered (Fig. 16a) [13], [66], [67]. Analyzing the plate

from figure 16b with initial geometrical imperfections, the following relationships could be written:

2h i

i1

;

2hi

(1a,b)

])2()

2[()h2( 2/1i2/1i2/1

(1c)

Using the principals of rigid-plastic analysis, the internal potential energy for the mechanism with

geometrical imperfections is written as follows [13]:

])2()

2)[(

cos

11()

tan

2(h

2

tf2U 2/1i2/1i

2

2/12

y

(2)

Taking into account the moment acting on the plate and the corresponding rotation:

2/1i

2

y

)2

(

1)(k

2

thf2

d

dUM

(3)

2/1ip )2

(

1)(k

h

t2

M

M

(4a)

)cos

11()

tan

2()(k

2

2/1

(4b)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

After a minimization process, the value of the k(α) factor results for α = 0.6847 which corresponds to

an angle of 39.29 degrees. The obtained value is similar to that obtained by Park and Lee [66].

Therefore, for the k = 4.1736 the relationship (4) gives:

2/1ip )(

1

h

t347.8

M

M

(5)

Plotting the equation (5) for instance applying h/t = 40 and θi = 0.0-0.03, it is showed in figure 16c

that geometrical imperfections, at the level of M=Mp, reduce progressively the rotation capacity.

Accordingly the rotation capacity results for the intersection of M/Mp = 1 with the curve given by the

equation (5), thus resulting a rotation capacity equal θ = 0.04355 rad.

In order to study the influence of pulsatory loading in Figure 17 the evaluation of rotation limit of a

bended plate can be done by means of four steps for a plastic rotation, θp = 0.04355/4 = 0.01089 rad,

each being composed by loading, unloading and reloading of a rigid-ideal plastic material. After the

first step, the remaining rotation is equal with θp. The second step starts with this initial rotation and at

the final stage, of this cycle, the remaining rotation is equal to 2θp. After the first two steps, the curve

corresponds to an initial plastic rotation of 2θp achieving the value of 0.02177 rad while a decreasing

of bending moment occurs. At the following step, with an initial rotation of 3θp = 0.03263 rad a new

decreasing is produced due to the increasing of initial rotation. Therefore, a new eroded postcritical

curve results, due to pulsatory loading, different from the one obtained under monotonic loading.

Hence, the concept of using geometrical imperfections (initial geometrical deformation) in order to

approach the softening behavior of a plate, under repetitive actions, seems to be compatible with real

behavior. The following conclusions yield from this bended plate model:

- The first two pulses produced in the range of M = Mp have no contribution to the reduction of

rotation capacity.

- The rotation capacity for the pulsatory loading is reduced to about half of the one corresponding

to monotonic loads.

5.3 Plastic collapse mechanism of I-shaped steel beams under cyclic loading

At the member level the plastic mechanism is introduced in the position where the plastic hinges

are intended to be formed (adjacent to the column face or at a distance from the column face as a

function of the joint detailing). Hence, this mechanism is a tool that permits to describe the inelastic

behavior of the beam that belongs to a frame. Under cyclic action the following response is observed

(Fig. 18a):

The positive moment, M+, causes buckling to the compressed upper flange while the section

rotates around the point O+ situated near the tensioned flange.

The negative moment, M-, causes buckling to the lower compressed flange while the section

rotates around the point O- located near the opposite flange.

Under the reversals of the seismic action the process continues in the same way also to the next

cycles. The buckled flanges and the rotation point are always situated in different position while the

tension strains are not able to straighten the buckled flanges. As a result the collapse mechanism is

developed by the superposition of two local plastic mechanisms. Moreover, during each next cycle the

element works with an initial geometrical imperfection as resulted from the previous cycle.

Consequently, an accumulation of plastic deformations in the buckled flanges, associated with the

number of cycles, could be observed (Fig. 18b). Physically the aforementioned process produces a

gradual degradation which creates the condition for a differentiation between monotonic and cyclic

loading effects.

With respect to the monotonic loading, the methodology of the local plastic mechanism was used

by many researchers for the prediction of the local ductility of steel members, but systematically has

been developed at the “Politehnica” University of Timisoara [3], [4], [13], [40], [65] (Fig. 18). The

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

local plastic mechanism is composed by yielding lines and plastic zones, which primarily dissipate the

input energy (Fig. 19). The proposed plastic collapse mechanisms, developed under monotonic

loading, further extended introducing the effect of cyclic bending which under cyclic action produces

accumulation of plastic deformations with a main consequence of gradual strength and deformation

deterioration. In case of cyclic actions a steel beam works under positive and negative bending

moments which, due to repetitive character, cause accumulation of plastic deformations. From

experimental observations and analytical works the following issues could be remarked:

- Under cyclic loading, during experimental tests a superposition of two plastic mechanisms for

positive and negative moments was clearly observed (Fig. 20a).

- For the first semi-cycle buckling occurs in a upper compressed flange while the section rotates

around a point in or near the opposite flange. The tensile forces in the opposite flange are very

small (Fig. 20b).

- For the reversal semi-cycle the bottom compressed flange buckles too, but due to the fact that

the tensile forces are small the reversal action is not able to straighten the upper buckled

flange.

- During the next cycles, nc+ (for upper flange) and nc

- (for bottom flange), the section works

with an initial geometrical deformation, Δi,, resulted from the previous cycle, nc+

-1 and nc--1,

respectively. Therefore, after each cycle an additional deformation is superimposed on the

previous one and in this way an accumulation of plastic deformations is achieved.

- The initial deformation increases with the increasing of cycles.

After the occurrence of flange local buckling a deviation from the characteristic monotonic

behavior is generally observed (Fig. 21) due to the fact that some yielding lines and plastic zones

became ineffective reducing the ultimate available rotation capacity. There are no differences between

monotonic and cyclic loading until the buckling of compressed flange. This observation allows for an

extension of the stable part of moment rotation curve to its unstable part. The difference in behavior

begins to be significant only after plastic buckling occurs at the cycle nb. It is evident that the level of

degradation is directly associated with the number of cycles. Thus, the accumulation of residual

deformations in flange and web deteriorates the load carrying capacity and reduce the rotation

capacity of a steel element.

Hence, the proposed model for cyclic loadings is based on the concept of accumulated initial

deformations, described by the shape of local plastic mechanism, in the same manner as initial

geometrical imperfections act in the elastic field for stability problems.

5.4 Local member plastic mechanism

Due to the uncertainties and inherent variability of earthquake ground motions it is difficult to

simulate the structural response without making a series of assumptions regarding the local inelastic

behavior which is defined as an envelope. Considering conceptually as θp the plastic rotation of a

hinge, θb the rotation at the level of flange buckling appearance and using the methodology of the

plastic collapse mechanism [3], [40] one can observe that the difference between monotonic and cyclic

mechanism is the introduction of the accumulated plastic rotation, θac:

- for monotonic loading

2/1)M(2)M(1

p

1

M

M

; (6a)

- for cyclic loading

2/1

)ac

)M(2)M(1p (

1

M

M

(6b)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

where α1(Μ) and α2(Μ) constants function on geometrical dimensions of cross-section, determined in

[13], [40], [68].

It is clearly evidenced that the erosion of rotation capacity is related to the number of cycles. As it

was shown not all the cycles produce an erosion of member ductility. Therefore, if it considered the

maximum number of pulses proposed in section 3.2, one can define the following cycle types:

- n, total number of cycles;

- nb, number of cycles produced in stable elasto-plastic field, before the reduction of moment

capacity due to local buckling;

- nc, number of decisive strong pulses producing the erosion of rotation capacity:

nc = n - nb (7)

Hence, the cyclic ultimate rotation, θuc, could be resulted from the monotonic one, θum:

θuc = θum – θac (8)

where (Fig. 12):

-for constant loading:

θac= (nc – 1) θp (9a)

-for increasing loading:

θac = nc (nc – 1) θp / 2 (9b)

In terms of the rotation capacity, considering the ECCS loading history [24] that recommends the step

for each cycle to be θp, one can obtain the following relationships:

- cyclic rotation capacity under constant amplitude:

Rcyclic.c = Rmon - (nc – 1), with nc ≥ 2 (10a)

- cyclic rotation capacity under increasing amplitude:

Rcyclic.in = Rmon - nc (nc – 1) / 2, with nc ≥ 2 (10b)

where Rmon is the monotonic rotation capacity determined as (Fig. 22) [3], [4], [40]:

Rmon = (θu /θp)-1 (11)

As a result of the conventional step for cyclic loading proposed in [24], the number of cycles

produced in stable elasto-plastic field, and considering that the maximum moment Mb corresponding

to the rotation θb occurs at ~ 0.7 θu (Fig. 22), yields:

nb ~ 0.7 Rmon (12)

Examining the relationship (7), on can see that there are two different cases:

- if n < nb , nc < 0, all the cycles occur in the stable part, before the plastic bucking of the

compression flange, and as a matter of fact the cyclic loading has no effect on the determined

monotonic rotation capacity;

- if n > nb, nc > 0, a reduction of rotation capacity will occur, then a correction of the monotonic

rotation capacity, due to the effect of cyclic loading, must be necessary.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

The relationship (12), estimates the cycle number through the stable field providing a prediction

regarding the maximum number of cycles for an expected earthquake. Moreover offering the

possibility to establishing the required Rmon in the way to eliminate or to minimize the damaging

effects of cyclic loading. For instance, for an earthquake for which the expected effective cycle

number is 6, it is possible to choose a profile having Rmon = 6/0.7 ~ 8.5, for which the cycle loading

has no any reduction, or a profile having Rcyclic ~ 5.5 with high cyclic ductility (see Section 6.3),

The definition of the monotonic rotation capacity is conventionally related to fully plastic moment,

in the lowering post-critical curve at the intersection with the theoretical plastic moment (Fig. 22).

However, for the cases of cyclic loading, in order to determine the monotonic rotation capacity which

will be further corrected, the intersection of lowering post-critical curve with the value 0.9 Mp is

proposed to be used [13].

6. AVAILABLE BEAM DUCTILITY FOR FAR-SOURCE EARTHQUAKES

6.1 DUCTROT-M computer program used for the prediction of cyclic available ductility

The DUCTROT-M computer program (free downloadable [69]) is developed with the aim of

determining the rotation capacity of steel members, validated by the verification with experimental

and numerical results [3], [4], [13], [40], [68], [69]. Beyond the prediction of the rotation capacity

under monotonic load, DUCTROT-M software contains a series of functions for the calculation of the

rotation capacity under cyclic load. Based on the above mentioned methodology, in the following

some further relationships are presented giving a holistic view of the program.

The erosion of the ultimate rotation can be evaluated from the following equations:

-for constant rotation amplitude:

pu i (13)

where i is equal to nc or less than this value (e.g. the greatest value for which θuj-Δθu ≥ 0).

-for increasing rotation amplitudes:

pu2

)1i(i

(14)

where i is equal to nc or less than this value (e.g. the greatest value for which θuj-Δθu ≥ 0).

Finally, the ultimate or the rotation capacity as given by the program is:

unujujc (15a)

1Rp

ujccyclic

(15b)

Regarding fracture rotation, the software also facilitates the calculation of that one taking into

account the yield ratio, the form of the plastic mechanism as well as the sectional and member

characteristics.

6.2 Factors affecting the cyclic available rotation capacity

With the aid of the DUCTROT-M computer program an analysis was performed focusing on the

main influence parameters which affect the cyclic available rotation capacity. In this study European I-

sections, IPE and HEA widely applied as beams, was used.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

6.2.1 Loading type

The effect of cyclic loading conditions, with constant or increased amplitude, on the rotation

capacity is presented in Figure 23. For the case of increasing amplitude, taking into account the

number of cycles that produce a gradual deterioration, it is pointed out that after several cycles the

rotation capacity was exhausted as compared with the monotonic one. Also for constant amplitude,

one can observe a dramatic erosion of the available rotation capacity that approach more than 50% of

the monotonic one. Due to great uncertainties with regard to the ground motion the available ductility

could be determined as an envelope which is defined by an upper bound given by the increasing

amplitude and a lower bound described by the constant amplitude.

Looking from a different point of view and assuming that the increasing loading type is directly

related with soft soil conditions while the constant one with stiff soil conditions, it could be remarked

that the same member has a different available inelastic rotation capacity due to a different process of

the accumulation of plastic deformations. Using this concept the effect of local soil conditions on the

member available ductility is achieved. Therefore, as a function of earthquake type (for instance far-

source seismic action) and local soil conditions not only the required ductility is influenced but also

the available one.

6.2.2 Cross section conformation

The influence of section type as a function of cycles producing erosion after flange buckling is

plotted in Figure. 24. Comparing two different types of hot-rolled sections with approximately the

same flange slenderness ratios (i.e. for IPE, c/tf = 5.11, for HEA450, c/tf = 5.85) and different web

slenderness (i.e. for IPE, d/tw = 38.48, for HEA450, d/tw = 29.91) as well as with different load

carrying capacity, one can observe that HEA 450 keeps constant the cyclic rotation capacity, which is

the same with monotonic one, while for IPE a gradual degradation of the cyclic rotation capacity is

finally remarked. Using the DUCTROT-M computer program an extensive analysis for the hole range

of HEA and IPE was carried out revealing the same conclusion. HEA sections have a superior

behavior due to greater web thickness and lower web height and greater intersection zone between

flange and web, [4]. In order to demonstrate the effect of the two aforementioned critical parameters a

sensitivity analysis was carried out using an IPE section varying the web thickness and height while

keeping constant the flange width and thickness. From figures 25a,b one can observe that the

increasing of web thickness leads to the increasing of rotation capacity, while the reducing of web

height also leads to the increasing of the rotation capacity. However, the effect of web thickness is

more pronounced than the web height; thicker web plates increase the number of cycles until buckling,

nb. A certain web thickness provides the possibility for a stable deformation of the flange supported by

the web. Moreover, identical remarks were pointed out from the experimental work published in

[36],[37],[38],[39].

Therefore, for design purposes the following recommendations are proposed:

The use of HEA sections for the conformation of moment resisting frames supported on soft

soils.

A structural detail improving the inelastic capacity of a plastic hinge as presented in Figure 26.

According to column tree concept, the stub could be shop welded from HEA section and the

remaining part of the beam from IPE. In this manner, in a conventional way, both economical

and behavioral requirements could be achieved. In addition, the HEA section could be

fabricated as a reduced beam section, which is more advantageous than IPE section due to

better adjustments of flange reduction, as a function of plastic capacity, as well as easier

fabrication further enhancing the rotational capacity of steel beam.

6.2.3 Yield to Ultimate stress ratio

The investigation of the influence of yielding ratio, ρy = fy/fu, on the cyclic available ductility is

directly related to the number of cycles, nc, producing erosion of the local available ductility. Figures

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

27a,b plotted for IPE sections, shows the accumulation effect, introduced by a certain number of

cycles, comparatively with the variation of yielding ratio; however, one can observe the following

three main issues. Firstly, the effect of cyclic cumulative action has a more pronounced character than

the increasing of yield to ultimate stress ratio. Secondly, the effect of the aforementioned ratio

becomes noticeable for values greater than 0.80 where the reduction is approximately 20%-30% as

compared with the one of a ratio equal to 0.65 and as a function of loading type (increasing vs.

constant amplitude). Finally, the reduction is more severe in case of increasing amplitude than in case

of constant one, but in any case the dramatic reduction is due to number of cycles producing ductility

degradation. For HEA sections and for low values of ρy equal to 0.65 the effect of cyclic action

disappears, however with the increasing of ρy in the range 0.75-0.95 the rotation capacity is reduced

due to the predominately effect of cyclic deterioration, Fig 28a,b. The divergence in the behavior of

HEA and IPE sections is because of different sectional conformation (e.g. web and flange width-to-

thickness ratio).

Current Eurocode 3 [2] regarding the yielding ratio prescribe a limit value of 0.90 (in the code the

ratio presented inverse as fu/fy >1.1). The previous mentioned limit seems to be proper for static

conditions, while for cyclic and especially for strain-rate conditions is unsuitable [15]. A proposal, for

cyclic far-field actions, to limit the yielding ratio to a value of 0.85 (or in the code format a value of

1.20) is considered to be convenient taking into account the capabilities of producers. Moreover,

Eurocode 8 specifies only measures regarding the assurance of yielding strength, fy (clauses 6.2 and

6.11). In this direction, in order to fill the gap for material control it is necessary the introduction of

such a ratio, creating more stable conditions for the development of the capacity design.

Accordingly, an experimental and analytical work presented in [27],[70],[71], concludes that for a

ratio of 0.85-0.95 the rotation capacity is reduced under the value of 0.03 rad which is considered as a

benchmark value. However, due to a great number of factors affecting the local ductility (e.g.

detailing, loading, workmanship, etc), the difficulty and the differentiation in experimental conditions

to capture the critical deformations, a level of conservatism should be considered.

After an extensive parametrical study it is observed that the conclusions previously mentioned are

valid for the whole range of IPE and HEA sections. It is a very important remark when one tries to

associate the monotonic and the cyclic local ductility with the ductility classification. Hence, for

design conditions within the range of material variability between ρy = 0.65-0.80, the influence of

yielding ratio does not affect the cyclic available rotation capacity; the number of cycles being of

primary importance. For values larger than 0.85, it is necessary for the designer to proceed in the

prediction of the cyclic rotation capacity further reducing the rotation due to the material variability.

6.2.4 Yield strength limit

Dealing with the influence of yield strength, at a first glance, appears to be decisive regarding the

rotation capacity under cyclic action, Fig 29. Nevertheless, the severe ductility reduction is associated

with the number of cycles as compared with the influence of the increasing yielding strength,

especially for IPE sections. Moreover, the same is true for the HEA sections made by an S355 steel

quality. For lower qualities of HEA sections the effect of cycles seems to be ineffective. In fact the

stable behavior of these type of cross-section derives from the cross section conformation (Fig. 24,

25). Experimental work carried out by Calado & Azevedo [64] reveal the same conclusion.

As a conclusion from both paragraphs, 6.2.3 and 6.2.4, it could be pointed out that the influence of

material variability and steel quality is covered by the effective action of a certain number of cycles.

6.2.5 Strength degradation

In order to study the strength degradation of hot rolled profiles the loading history proposed from

ECCS was applied in combination with the concept of the cumulated initial deformation presented in

this paper. Figure 30 shows comparatively the gradual strength degradation of I shaped beams

conformed by IPE and HEA sections. It is clear that HEA sections have a better strength capacity than

IPE sections with about 30%-35% as revealed by a further parametrical analysis. The superiority of

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

HEA sections was also provided by the experimental work presented in [36],[38],[64]. Furthermore,

the experimental findings are also validate the concept of the initial cumulated deformation.

Therefore, taking into account all the aforementioned influences that affects the cyclic available

ductility appears to be in the safe side the use of HEA sections for the conformation of moment

resisting frames.

6.3 Classification of cyclic available member ductility

In case of cyclic action the classification could be associated with the monotonic one taking into

account the main effect producing strength and ductility degradation which is the number of cycles

producing erosion of the monotonic ductility after local buckling. Moreover, in [3] and [4] was

pointed out the deficient framework for the prediction of the available ductility under monotonic

loading. In this direction a new classification was proposed based on the member level of inelastic

behavior. The numerical analysis was revealed that a reduction of 2 to 4 as compared with the

monotonic determined ductility is the effect of cyclic action of IPE and HEA steel beams. Particularly

3 to 4 for IPE sections and 2-3 for HEA sections (Fig. 23, 24, 26, 27). Consequently, results that Rcyclic

= ccyclic Rmonotonic, where the correction factor ccyclic = 1/nc taking values between 0.25-0.50 as a

function of member conformation.

Accordingly, exploiting the limits proposed in [3], making use of the basic relation for the rotation

capacity, R = (θu /θp)-1 and furthermore applying a correction factor γc (with a mean number of

effective cycles producing degradation equal of nc = 3), the following Cycle Ductility Classes, at a

member level of classification is proposed:

High Cyclic Ductility, HCD ≥ 2.50

Medium Cyclic Ductility, 1.50 ≤ MCD < 2.50

Low Cyclic Ductility, LCD ≤ 0.50

In fact similar values were obtained from experimental studies [30], [34]. However considering, for

instance, the target value of 0.035 rad, specified in [1] for high ductility class performance, and

applying for some IPE and HEA sections also values of the same order were obtained, Table 2.

Therefore, the previously mentioned limits appear to be reasonable for practical design. In this manner

taking the results from [4] and applying a reduction factor of 4 for IPE sections and 3 for HEA

sections, in Tables 3,4, the evaluation of cyclic member ductility are presented. Ductility of HEA

sections remain unchanged while for IPE sections a degradation of one class is observed. In any case,

the section classification specified in Eurocodes [1],[2] seems to be ineffective also for cyclic

conditions as was demonstrated also for monotonic conditions [4].

7 CONCLUSIONS

This work, taking into account both the engineering seismology as well as the structural aspect,

studies the framework for the prediction of the available member ductility of steel I-shaped (IPE and

HEA) beams under far-source earthquake ground motions. Based on the methodology of the plastic

collapse mechanism, implementing the concepts of ultra low cycle fatigue and the initial cumulative

deformation, a numerical study was performed. By using the DUCTROT-M computer program a

parametrical analysis was carried out in order to evaluate the main influencing parameters affecting

the rotation capacity under far-source loading conditions. Such factors are the loading type, the

number of cycles producing strength and ductility deterioration, steel quality and the yielding ratio.

More specifically the main conclusions of this study are summarized as follows:

An erosion of the available local ductility as compared with the monotonic one and in function

of loading history, site conditions (introduced by the constant amplitude for normal soil

conditions and increasing for bad soil conditions), and number of cycles is observed. The

degradation is more severe in case of increasing amplitude, where the effective number of

cycles withstanding inelastic deformation is also reduced.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Taking into account that a certain number of cycles, nb, occurs before the formation of the

plastic local buckling, the proposed methodology provides the possibility to choose an I-

section for which the influence of cyclic loading is reduced.

With respect to the strength degradation, a more pronounced deterioration for IPE sections

than the HEA hot rolled sections is observed. Accordingly, the prediction of the cyclic

ductility is directly related with the number of cycles developing large plastic deformations,

nc. Hence, in order to obtain the cyclic available member ductility in function of the

monotonic one a reduction by a factor, nc, of 3 to 4, for IPE sections, and 2 to 3, for HEA

sections, is reasonable to be considered.

The member conformation strongly influences the cyclic behavior; however HEA sections

presents a more stable behavior than IPE sections, the first one being recommended to frame

structures especially in case of soft soil conditions. Generally, IPE sections are suggested to be

used for secondary elements or in gravitational loading conditions, not in such seismic

conditions and particularly in cases where the out of plane buckling is the decisive factor for

the failure.

The steel quality and the yielding ratio have a secondary detrimental effect on the cyclic

available ductility, mainly being covered by the effect of number of cycles producing

degradation due to an accumulated action of local buckling.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

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Tables Caption

Table 1. Selective cyclic experimental tests on steel beams.

Table 2. Calculation of rotation capacity for a target value prescribed in Eurocode 8 [1].

Table 3. Comparative presentation of member classification for IPE under monotonic and cyclic

loading.

Table 4. Comparative presentation of member classification for HEA under monotonic and cyclic

loading.

Tables

Table 1. Selective cyclic experimental tests on steel beams. Num. Author No

Tests

Section

Type

Test

Scheme

Loading

History

Slenderness Main Obs.

FS WS LS

1 Bertero & Popov [29] 11 RS MG CD 6,35 1,24 -

Revealed the main parameters of

cyclic loading (controlled

deformations, number of cycles

to failure)

2 Takanashi [30] 16 RS MG CD - - 51,20..80,90 Local flange buckling associated

with web buckling and/or lateral

buckling produce strength-

stiffness degradation. Web

buckling is associated to strength

degradation

3 Van et al [32] 50 RS MG CD - - 30-60L

4 Mitani et al [33] 90 WS MG CD 6,2..16,0 158..640 25,0..40,0

5 Suzuki & Ono [34] - WS CM - - - -

6 Guruparan & Walpole [31] 6 WS/RS MG ID 8,2..8,5 53,7..80,0

- A lateral slenderness greater

than 60-70 reduce seriously the

deformation capacity

7 Lee & Lee [35] 8 RS CM CD 9,6 30,4 - 60% deterioration in case of

cyclic action

8 Ballio & Calado [36], [37] 4 WS MG CD 10...25 38.3..60.5 - Once the flange has buckled the

maximum load cannot be

reached 9 Castiglioni [75] 14 RS MG ID 7..10 19,8..33,2

-

10 Ballio & Castiglioni [59]

[73],[74] 45 RS MG CD/RD 6,9..10,0 19,8..39,2

- Web slenderness is of crucial

importance for a stable cyclic

deformational capacity

11 Takanashi & Udagawa [41] 10 WS MG CD 15 35

61 Slab effect reducing the

deformation capacity and

increasing the stress level

13 Valente & Castiglioni [43] Concrete slab reduce the low

cycle fatigue strength

13 Green et al [44], [45] 13 WS MG ID 5,9..9,0 27,9..29,3 - Influence of yielding ratio of

more 0.85

14 Jiao et al [46] 4 WS MG ID 7,69 50 - Effect of Baushinger on plastic

capacity

15 Gioncu et al [47] 6 WS CM ID 9,0..13,0 60,0..71,0 - Superposition of plastic

mechanism in cyclic loading

16 Mateescu & Gioncu [48] 11 WS MG ID/PD 10,9..12,3 24,0..26,3 - Early fractures in case of

impulsive loading conditions

MG-moment gradient, CM- constant moment.

WS-welded sections, RS- hot-rolled sections.

CD-constant displacement, ID-increasing displacement, RD-random displacement, PD-pulse displacement.

Table 2. Calculation of rotation capacity for a target value prescribed in Eurocode 8 [1].

Section1) Span

[mm]

Target

ultimate

rotation, θu,

[rad]

Plastic

rotation, θp

[rad]

Rotation

capacity,R

Classifi

cation

IPE 300

6000 0.035

0.015 1.33 MCD

IPE 400 0.012 1.92 MCD

HEA 300 0.012 1.76 MCD

HEA 450 0.008 3.37 HCD 1)

Steel quality S235

R = (θu/θp)-1

θp = MpL/4EI

Table 3. Comparative presentation of member classification for IPE under monotonic and cyclic loading.

Steel

section

Ductility

type1)

L = 2000mm

L = 3000mm

L = 4000mm

L = 5000mm

S235

S355

S235

S355

S235

S355

S235

S355

IPE

300

Monotonic HD HD HD MD ---2)

--- --- ---

Cyclic HCD MCD MCD MCD --- --- --- ---

IPE

330 Monotonic HD HD HD MD HD MD --- ---

Cyclic HCD MCD MCD MCD MCD MCD --- ---

IPE

360

Monotonic HD HD HD HD HD MD MD LD

Cyclic HCD MCD MCD MCD MCD MCD MCD LCD

IPE

400

Monotonic HD HD HD HD HD MD MD LD

Cyclic HCD HCD MCD MCD MCD MCD MCD LCD

IPE

450

Monotonic --- --- HD HD HD MD MD MD

Cyclic --- --- HCD MCD MCD MCD MCD LCD

IPE

500

Monotonic --- --- HD HD HD MD MD MD

Cyclic --- --- HCD MCD MCD MCD MCD LCD

IPE

550

Monotonic --- --- --- --- HD HD HD MD

Cyclic --- --- --- --- MCD MCD MCD MCD

IPE

600

Monotonic --- --- --- --- --- --- HD MD

Cyclic --- --- --- --- --- --- LCD LCD

Legend 1)

Rotation capacity for in plane post elastic buckling mechanism. 2)

--- - Sizing of the member would be other than ductility limit state. For instance

serviceability limit state would be the predominant criteria for member sizing. 3)

L-standard beam span, [13].

Table 4. Comparative presentation of member classification for HEA under monotonic and cyclic loading.

Steel

section Ductility type

1)

L=4000mm L=5000mm

S235

S355

S235

S355

HEA

320 Monotonic = Cyclic HD/HCD MD/MCD HD/HDC MD/MCD

HEA

340 Monotonic = Cyclic HD/ HCD HD/HDC HD/HDC MD/MCD

HEA

360 Monotonic = Cyclic HD/ HCD HD/ HCD HD/ HCD HD/ HCD

HEA

400 Monotonic = Cyclic HD/ HCD HD/ HCD HD/ HCD HD/ HCD

HEA

450 Monotonic = Cyclic HD/ HCD HD/ HCD HD/ HCD HD/ HCD

HEA

500 Monotonic = Cyclic HD/ HCD HD/ HCD HD/ HCD HD/ HCD

HEA

550 Monotonic = Cyclic HD/ HCD HD/ HCD HD/ HCD HD/ HCD

HEA

600 Monotonic = Cyclic HD/ HCD HD/ HCD HD/ HCD HD/ HCD

Legend 1)

Rotation capacity for in plane post elastic buckling mechanism. 3)

L-standard beam span, [13].

Figure Caption

Figure 1. Generation of earthquakes in function of source type and depth.

Figure 2. Earthquake types with respect to the affected area.

Figure 3. Earthquake types in function of the epicentral distance (after [13]).

Figure 4. Influence of site conditions (after [13]).

Figure 5. Crustal earthquakes with shallow sources (after [13]).

Figure 6. Ground motions in function of the source-site distances (after [13]).

Figure 7. Sub-crustal earthquakes with deep sources (after [16]).

Figure 8. Duration effect in function of the epicentral distance.

Figure 9. Displacement time history in function of ground motion and structural type (after [22]).

Figure 10. Relationship between number of effective cycles and source-site distance (after [22]).

(a) Input energy dissipated by a section of structure;(b) Effective cycles in function of the epicentral

distance.

Figure 11. Conceptual behavior of a plastic hinge in case of far-field earthquakes.

Figure 12. Cycle loading types: (a) Constant amplitude; (b) Increasing amplitude; (c) Gradually

increasing amplitude.

Figure 13. Effects of cyclic loading: (a) Required ductility level; (b) Available ductility level.

Figure 14. Classification of fatigue (after [25]).

Figure 15 Accumulation of plastic deformations: (a) Shape of a plastic buckled section during an

experimental test; (b) Plastic mechanism for compressed flange for cycle loading.

Figure 16. Yielding mechanism of a bended plate: (a) Mechanism configuration; (b) Mechanism with

initial deformations; (c) Moment-rotation curve relationship, influence of initial deformations.

Figure 17. Moment-rotation curve of a cyclically bended plate with initial cumulative deformation.

Figure 18. Plastic collapse mechanism at the member level: (a) Positions of plastic hinges;

(b) Accumulated plastic deformation of a plastic collapse mechanism relative to the number of cycles.

Figure 19. Plastic local mechanism for monotonic loading.

Figure 20. Plastic local mechanism for cycle loading: (a) Experimental evidence; (b) Accumulation of

plastic deformations for buckled flanges.

Figure 21. Moment-rotation curve eroded by cyclic loading.

Figure 22. Determining the rotation capacity for monotonic loading .

Figure 23. Influence of the loading type on the available cyclic member ductility.

Figure 24. Influence of the cross-section conformation on the available cyclic member ductility.

Figure 25. Influence of web slenderness on the available ductility: (a) Effect of web thickness

(b) Effect of web height.

Figure 26. Structural detail enhancing the ductile capacity of a member that belongs to a frame.

Figure 27. Influence of the yielding ratio on the rotation capacity of IPE beams: (a) Constant

amplitude; (b) Increasing amplitude.

Figure 28. a) Influence of the yielding ratio on the rotation capacity of HEA beams: (a) Constant

amplitude; (b) Increasing amplitude.

Figure 29. Influence of steel quality on the available cyclic member ductility.

Figure 30. Moment-rotation curves according to ECCS Recommendations for loading history.

Figures

Figure 1

Figure 2

Figure 3

Figure 4

Figure 5

Figure 6

Figure 7

Far-field earthquake

Source

Epicentre

Deep source

Site conditions

Figure 8

Duration (sec) FAR-FIELD EARTHQUAKES

Figure 9

(a)

(b)

Figure 10

θ

(b)

(c)

Figure 12

Mp

(b)

Figure 13

Figure 14

(a)

Initial deformation

(b)

Figure 15

Plastic mechanism for

- cycle i

- cycle i – 1

- monotonic

loading

(a) (b)

(c)

Figure 16

(a)

(b)

Figure 18

Figure 19

(a)

Figure 20

Figure 21

Figure 22

Figure 23

Figure 24

a)

b)

Figure 25

Figure 26

a)

b)

Figure 27

a)

b)

Figure 28

Figure 29

(a)

(b)

Figure 30