Chapter 2: Deteriorating hysteretic models

Chapter 2: Deteriorating hysteretic models

2.1 Abstract

Within the international scientific community there is complete agreement about the importance of explicit modeling of beam-to-column connections in order to have a realistic and reliable prediction of the global behavior of steel frames, under both static and dynamic external actions. In fact column web panel zone may significantly affect stiffness, strength and ductility of the whole frame. There is also good agreement as the fact that inter-storey drift angle , the ratio between the displacement of one floor relative to the adjacent lower one and the storey height , could be considered the most significant parameter for measuring damage of the frame under seismic actions. On the contrary, there is no agreement regarding the correct methodology for evaluating the seismic performance of steel frames, which is a more general problem, involving both seismological and structural aspects.

From the structural point of view, one major uncertainty is related to the type of hysteresis model to be used for performing numerical analyses. In fact, looking at experimental results, it’s clear that usually assumed elastic-perfectly-plastic hysteresis model is far from being faithfully representative of the actual cyclic response of connections.

2.2 Della Corte et al. hysteretic model

2.2.1 Introduction

A synthetic picture of the frame seismic performance can be obtained by using the ‘performance curve’. This relates the ground motion damage potential , where is the spectral elastic pseudo-acceleration computed with reference to a 5% viscous damping ratio and for the first vibration mode of the structures, and the maximum damage (maximum transient value of the ratio) produced at that level of earthquake intensity. This curve is obtained by scaling a given ground acceleration time-history at increasing values of the peak ground acceleration (PGA), and computing the relevant maximum inter-storey drift angle. This procedure is called ‘Incremental Dynamic Analysis’ (IDA) or ‘Dynamic Push-over’ (DP), to contrast with the well-known ‘Static Push-over’ (SP). In the SP, lateral displacements are increased using a pre-fixed static pattern of increasing lateral forces, whilst in the DP they are pushed over using a pre-fixed acceleration time history scaled at increasing values of PGA. In order setting limiting values of inter-storey drift angles, in relation to the selected Limit State, it’s possible to evaluate the corresponding values of the e.q. intensity to be compared with the one given by the code, according to the limit state and the site seismic hazard.

Fig. 2.1 shows a qualitative picture of the ‘performance curve’ basing on the IDA analysis methodology. Some usual limits of inter-storey drift angles are provided, excepting for the limiting value for the structural stability performance level, which is not considered by the seismic codes. In Fig. 1, three horizontal dashed lines are also shown, indicating three levels of earthquake intensities, which correspond to the achievement of the serviceability , life-safety and structural stability performance levels, respectively. If the point of the performance curve, related to the generic performance level, is above the horizontal line corresponding to the reference value of spectral acceleration, then the performance objective has been achieved.

Fig. 2.1: a qualitative picture of ‘performance curve’ [5]

It’s useful to remark that if the two parameters chosen as representative of seismic damage potential and of actual frame damage (maximum ) are the correct ones, a monotonically increasing relationship should be expected. On the contrary, sometimes, an increase in the normalized spectral acceleration produces a constant (or even decreasing) maximum transient inter-storey drift. So is to be concluded that, in such cases, the chosen parameters are not completely representative. Nonetheless the ‘performance curve’, as defined above, is a useful tool which allows to quantify the influence of both the hysteresis model and the seismic input on the seismic response of the frame.

2.2.2 Della Corte et al. Model

The model developed by G. Della Corte et. al. [5] gives particular attention on the cyclic local buckling behavior. Cyclic buckling phenomena, occurring within the length of plastic hinges, induce strength degradation, accompanied by softening branches in the moment-rotation relationship. Actually an accurate simulation of the hysteresis behavior in this case is quite difficult, because the point of initiation of the softening branch moves during the deformation history. That’s why a simplified approach has been undertaken. It’s assumed that, during the deformation history, the rotation stiffness becomes negative (softening) when the rotation excursion in one direction becomes equal to the value of the rotation producing local softening in the case of monotonic loading. After it first activation, the softening branch remain fixed, thus defining a boundary for the moment-rotation curve (Fig. 2.2). The negative-stiffness branch is activated when the monotonic ‘softening rotation’ is achieved. The reduced flexural strength is set equal to the smallest value of the bending moment reached during the preceding deformation history along the softening branch, in the same direction of loading. Such a model is deemed to be adequate at least in the case of a small number of plastic excursion of large amplitude.

However, in order to simulate the cyclic strength degradation effect, another type of degradation rule is considered and superimposed with the one related to the reaching of excessive plastic excursions in one direction. Such a cyclic degradation law is based on the use of a ‘damage factor’ β.

Where M0 is the initial (undamaged) plastic strength of the connection, is the ultimate rotation for monotonic loading conditions, and is the dissipated hysteretic energy. According to this quantities, the reduced values of plastic strength is assumed as:

Fig. 2.2: reduction of bending strength owning to the activation of negative stiffness branches in the moment-rotation relationship [5]

Fig. 2.3 shows the modeling of pinching of hysteresis loop. Two limiting curves, the ‘lower bound’ and the ‘upper bound’ are introduced. Both curves are described using the formula proposed by Richard and Abbott [13], which is based on four parameters: initial stiffness k0,

strength F0, hardening stiffness kh, and shape factor n.

Fig. 2.3: simulation of pinching [5]

The generic point, which is interpreted as a transition from the lower to the upper bound, is obtained again using Richard and Abbot formula, where the generic of the above parameters (pi) is computed by following linear combination:

In which pil and piu are the ith parameters to the lower-bound and the upper-bound curves, and

pit are the parameter to the transition curve. The parameter t defines the shape of the

transition and has been assigned the mathematical low:

Where t1, t2, φlim are additional model parameter.

2.2.3 Study case

Several numerical analyses can be carried out with reference to a regular plane frame of a steel building. Important inputs of the design procedure are:

 Used Code  Sub-soil class  PGA

 Design q-factor

 Limits of the inter-storey drift capacity at different Limit States

Beam-to-column joints can be assumed to be full-strength (rigid) or partial-strength (semi-rigid). Fig. 2.4 shows the geometry of an exemplified model. Column-web panel zones have been assumed to be always rigid and resistant enough to remain elastic, and the reduction of the flexural length of beams and columns due to the finite size of joints has been neglected. Beam-to-column connections have been modeled by means of lumped elasto-plastic springs. Two types of connections can be considered:

I. Rigid and full-strength connections characterized by moment-rotation relationships with strength degradation.

II. Semi-rigid and partial-strength connections characterized by moment-rotation relationship with or without pinching of hysteresis loops.

Fig. 2.4: beam-to-column joint models [5]

2.2.4 Seismic Input

It’s important, for performing numerical analysis, to select different ground acceleration time-histories which have to be representative respectively of the near-fault recordings and of the far-fault ones. Different values of ‘Trifunac duration’ [12] are expected. This criterion of subdivision reflects emphasis on the effect of the earthquake duration on the seismic performance when strength degradation is taken into account. In fact, it’s apparent that simulating cyclic strength degradation might be more important for long duration earthquakes than for pulse-type ones.

2.2.5 Analysis

In the analysis of the structural response, the acceleration records has to be scaled with PGA, starting from a rate of PGAr (recorded PGA), and with a defined increasing step. The generic ‘performance curve’ (elastic spectral pseudo-acceleration at 5% viscous damping and 1st _period of elastic vibration vs maximum inter-storey drift angle) has to be terminated when the dynamic instability occurs. This condition is deemed to be representative of the frame ‘collapse’, i.e. the achievement of a limit situation where the structure’s capacity of bearing vertical loads is lost or unacceptably reduced. However sometimes, more frequently when non-degrading hysteresis models are adopted, the ‘performance curve’ still increases for very large values of inter-storey drift angle and/or spectral acceleration. In those cases the analysis have to be stopped when a very large inter-storey drift angle and/or spectral acceleration is achieved.

2.2.6 Analysis – continuous frame with strength degrading beam-to-column connections

Fig. 2.5 shows an example of the influence on the computed seismic performance of the strength degradation produced by local softening phenomena, i.e. by the attainment of plastic deformation excursions larger than the ones leading to the activation of the negative rotational stiffness branch. Two different values of ‘softening rotations’ are taken into account.

Fig. 2.5: effects of local softening phenomena on the seismic performance of the whole frame [5]

Fig. 2.6 illustrates the influence of cyclic strength degradation, and the modification of the frame ‘performance curve’ with increasing values of the ‘damage factor’ . In particular indicates a very low level of strength degradation, while indicates a quite strong strength degradation.

Fig. 2.6: effect of cyclic fatigue-related strength degradation phenomena on the seismic performance of the whole frame [5]

In both figures is clearly understandable that the ‘performance curve’, for low levels of drift demand, is nearly insensitive to local softening phenomena and to the level of strength degradation. On the contrary the type of model becomes very important for large drift demands.

2.2.7 Analysis – semi-continuous frame with beam-to-column connections characterized by

pinching

In fig. 2.7 two types of hysteresis models have been considered, assuming the same skeleton curve but a different cyclic behavior (with and without pinching). Moreover the results presented are based on the adoption of non-degrading hysteresis models.

One can see, from the below diagrams, that is not possible to trace a general trend for the effects of pinching. In fact it’s deeply related of the to the type of ground motion.

Fig. 2.7: Effect of pinching of hysteresis loops on the seismic performance of the whole frame [5]

2.2.8 Conclusions

 Seismic evaluation of MR steel frames is usually carried out using simplified hysteresis models, like the elastic-perfectly-plastic one, together with the assumption of inter-storey drift angle capacity having upper limits over which strength degradation is likely to take place.

 In this way the ultimate limit state of the frame is practically identified with the attainment of the level of inter-storey drift angle for which the numerical validity of the model itself is lost.

 Numerical results can demonstrate that for relative low levels of inter-storey drift angle demands, the type of model used in the numerical analysis is of little significance.  If simplified model are used, the real safety at collapse remains unknown.

 Using the realistic hysteretic models there is a change towards a more physically meaningful ultimate limit state, that corresponds to the achievement of a situation where the building’s capacity for carrying vertical loads is lost or unacceptably reduced.  The use of the elastic spectral pseudo-acceleration (as the ground-motion damage

potential parameter) and the maximum transient value of the inter-storey drift angle (as the damage measure for the frame) gives rise to relationship which are not always monotonically increasing (as expected).

 Design of steel structures according to EC8 leads to strongly over-resistant structures, due to the limitation on inter-storey drift angles under frequent earthquakes (serviceability requirement). When judging the effect of degradation phenomena on the seismic performance of steel MR frames, this over-strengthening should be taken into account, since it reduces the impact of degradation itself.

2.3 Mele - Calado - De Luca hysteretic model

2.3.1 Introduction

The confidence of structural engineering in welded moment resisting frames (wMRF) was strongly compromised by the performances observed in the Northridge and Kobe earthquakes. Following these earthquakes, extensive unexpected brittle connection damage were detected in several frames. These brittle modes of failure had been defined “unexpected” since the wMRF connections were usually considered as the ones characterized by the more stable and ductile behavior, with large rotational capacity and energy dissipation. Starting with these observations, significant researches have been done, as the US, as in Europe and Asia, in order to enrich the experimental database and to assess the major parameters affecting the cyclic behavior of beam-to-column connections. In this paragraph an overview on the experimental program and results carried out on welded connection is reported, with the aim of defining the influence of the column size and the panel zone (PZ) design on the cyclic behavior.

2.3.2 The experimental program

2.3.2.1 Aim

Evaluating the effect of the column dimension and panel zone design on the cyclic behavior, ultimate strength and deformation capacity of welded connections, varying the applied leading history.

2.3.2.2 Specimen material and geometry

All the specimens, made by S235 steel, are double T shaped beam-column elements. The assembly consists in a 1000 mm long beam and a 1800 mm long column. Beam section is always the same (IPE 300), while three type of column section have been used (HEB 160, HEB 200, HEB 240). The three tests have been appointed, respectively, as BCC5, BCC6 and BCC8. Due to the relative cross-section dimensions of column and beam in the three series of connection specimens, the beam plastic modulus is respectively larger, approximately equal and smaller than the column plastic one.

The beam flanges have been connected to the column flange by complete joint penetration (CJP) groove welds, while fillet welds connect beam web and column flange. The continuity of the connection through the column has been ensured by horizontal 10 mm thick plate stiffeners, fillet welded to the column web and flanges.

2.3.2.3 Experimental set up, instrumentation plan and loading histories

The test set up, represented in Fig. 2.8a, consists in a foundation, a supporting girder, a reaction r.c. wall, a power jackscrew and a lateral frame. The power jackscrew (1000 kN capacity, +/- 400 mm stroke) attached to a specific frame, pre-stressed against the reaction wall and designed to accommodate the screw backward movement. The specimen is connected to the supporting girder by two steel elements.

Specimens are instrumented with electrical displacement transducer ( ), for carefully recording the various phenomena occurring during the tests. The typical instrumentation set-up is provided in Fig. 2.8b.

Fig. 2.8: a) specimen geometry b) specimen instrumentation [4]

The complete set of loading histories is provided in the following table, where loadings are defined in terms of: applied beam tip displacement ( ); applied beam tip displacement normalized to theoretical yield displacement ( ); inter-story drift angle ( ), i.e. normalized to the distance between beam tip and column centerline .

Column cross-section

HEB 160 HEB 200 HEB 240

A

B

BB

C Stepwise Increasing (ECCS) Stepwise Increasing (ECCS)

D Stepwise Increasing (ECCS)

E Monotonic Monotonic

Monotonic

2.3.3 Experimental results – global behavior and failure modes

In the following the experimental results are provided, with particular aim on the cyclic behavior and the failure modes observed for the three sets of specimens. In the Moment-rotation hysteresis loops, the Moment-rotation values have been calculated both as the “unprocessed” total rotation given by the applied inter-story drift angle , and as the beam rotation obtained through the measured LVDTs displacements at the beam cross sections. Correspondingly, in the experimental curves the moment is evaluated at the centerline of the column, while in the curve the moment is evaluated at the column face. In the figure below the beam plastic rotation has been obtained through the measured displacements at the transducer 1 and 2 (see Fig 2.9) by subtracting the contributions of the beam and column elastic rotations as well as the panel zone distortion.

Fig. 2.9: a) Moment-global rotation curves b) Moment-beam plastic rotation BCC5C, BCC6C and BCC8D tests and Moment-panel rotation curves [4]

2.3.4 Specimens BCC5 (HEB 160 column)

The cyclic behavior of the specimen BCC5 is characterized by a great regularity and stability of the hysteresis loops up to failure, with no deterioration of stiffness and strength properties. The very last cycle present a sudden and sharp reduction of strength, corresponding to the collapse, which occurred due to fracture initiated in the beam flange and propagated also in the web. During the test, significant distortion con the joint panel zone has been observed, while not remarkable plastic deformation occurred in the beam.

In the follow table a summary of the complete plastic cycles to collapse and the failure mode of the specimens is reported.

test N° cycles Failure mode

A 16 Crack on the beam flange close to the weld, propagated in the web

B 5 Fracture of the beam flange near the weld

BB 4 Crack on the beam flange close to the weld, propagated in the web

C 18 Crack on the beam flange close to the weld, propagated in the web

D 23 Fracture of the beam flange

2.3.5 Specimens BCC6 (HEB 200 column)

Two different kind of cyclic behavior have been observed for the BCC6 specimens. In some cases (C and D tests) it’s close to the behavior observed for the BCC5 type, with almost no deterioration up to the last cycle. On the contrary for the other tests (A, B, BB) a gradual reduction of the peak Moment at increasing number of cycles is evident. In this cases, from the very first cycles, local buckling of the beam flanges occurred, and a well defined plastic hinge has formed in the beam. The contribution of the panel zone deformation is not as significant as in the BCC5 specimen type.

In the follow table a summary of the complete plastic cycles to collapse and the failure mode of the specimens is reported.

test N° cycles Failure mode

A 15 Plastic hinge. Crack on the beam in the buckle zone, at 15 cm from _{the column face}

B 11 Plastic hinge. Crack on the beam in the buckle zone, at 15 cm from

the column face

BB 6 Plastic hinge. Crack on the beam close to the weld line, propagated _{in the beam web}

C 15 Less evident plastic hinge. Crack in the beam flange along the weld

line

D 18 Less evident plastic hinge. Crack in the beam flange, developed _{close to the weld}

2.3.6 Specimens BCC8 (HEB 240 column)

The hysteresis loops (excepting the one of the C test) show a gradual reduction of the peak moment starting from the second cycle, where the maximum value of the applied moment has been usually registered. This deterioration is related to occurrence of local buckling in the beam flanges an web. A well defined plastic hinge has formed in all the tests. The panel zone deformation is not remarkable.

In the follow table a summary of the complete plastic cycles to collapse and the failure mode of the specimens is reported.

test N° cycles Failure mode

A 12 Plastic hinge. Crack on the beam in the buckle zone, complete _{fracture of the beam flange and web}

B 16 Plastic hinge. Crack in the beam flange close to the weld, _{propagated in the beam web}

BB 2 Plastic hinge. Crack in the beam flange close to the weld , _{propagated in the beam web}

C 18 Plastic hinge. Crack in the beam flange at the buckled zone

D 15 Less evident plastic hinge. Crack in the beam flange, developed in _{proximity of the weld}

2.3.7 Comparison and observations

2.3.7.1 Panel zone and beam rotations

The contribution of the total (elastic + plastic) PZ deformation to the global rotation could be considered: remarkable in the BCC5 specimens, having the smallest column sections, less significant in the BCC6 specimens and minor in the BCC8 specimens, characterized by the largest column sections. The plastic rotations of the beams have been: minor for BCC5 specimens, comparable to the PZ rotations in the BCC6 specimens, larger for the BCC8 ones. The values of the total rotation capacity, which, in the increasing amplitude test, reaches 0.064 rad for the BCC5 specimen, 0.053 rad for the BCC6 and 0.046 rad for the BCC8 ones, correspond to low values of beam plastic rotations, respectively equal to 0.0057, 0.0175 and 0.0242 rad for the three specimens, thus confirming that large rotations can be experienced thanks to column web panel deformations.

2.3.7.2 Effect of column size on the cyclic behavior and failure mode

The BCC5 specimens, even though able to experience high deformation levels, have shown brittle failure modes in all the cyclic tests, with hysteresis loops practically overlaid and no degradation of the flexural strength up to the very last cycle, where a sudden decay of the carrying capacity occurred due to fracture, generally developed in the proximity of the weld. On the contrary the BCC8 specimens have exhibited a typical ductile behavior, with formation of a well defined plastic hinge in the beam starting from the first plastic cycles, and a gradual decrease of the peak moment at increasing number of cycles up to the collapse.

The BCC6 specimens displayed an intermediate behavior between the BCC5’s one and the BCC8’s one, depending on the applied loading sequence. With regard to the final collapse of the specimens, in the former cases it involved fracture in the beam starting at or close to the weld location, while in the latter cases it was due to the cracking in the buckled zones of the beam flanges.

2.3.7.3 Effect of the loading history

The different cyclic histories applied to the specimens have evidenced the dependence of the plastic deformation capacity on the loading histories. Also under the same history, different specimens have shown different behavior as well as the number of plastic cycles before failure and the failure modes.

2.3.8 Conclusive remarks

The quite high values of the maximum global rotations of welded connections obtained in these test can be related to the following aspects:

 The connection rotation capacity and ductility strongly decreases as the beam depth increase. Thus higher rotations are expected for beam-to-columns connections usually used in Europe, where the depth of the beam section (300-450 mm) is significantly less than the ones used in US practice (500-1000 mm), due to the current adoption of perimeter frames configuration.

 Fully welded connections have already shown, in several experimental tests, higher rotation capacity than the bolted web welded flange (BWWF) connections.

 A significant contribution of PZ deformation could suggest the possibility of utilizing the joint panel for providing energy dissipation and stable behavior of the connections even at large number of cycle.

2.4 Deng – Bursi - Zandonini hysteretic model

2.4.1 Abstract

A hysteretic connection element for the simulation of hysteretic response of connection is formulated to be implemented in a non-linear finite element program. In the formulation, both stiffness and strength degradation and pinching are expressed as functions of damage state variables. In this model damage is monitored and failure is detected by means of damage index. An automatic event definition algorithm is developed and an event-to-event solution scheme is followed. The element is applied to the simulation of hysteretic responses of unstiffened extended end plate connections, web panels, joints and frame substructures subject to cyclic reversal loading. For this model simulations are in good agreement with experimental data in term of loading-deformation relationship, strength, stiffness and energy dissipation.

Insofar as moment resisting steel frames are concerned, extensive investigations carried out in the last three decades have built up a satisfactory knowledge of the behavior of rigid beam-to-column joints under cyclic reversal loading. Recently, an alternative philosophy has been proposed for joint design, which is based on the use of semi-rigid beam-to-column connections. Under static loading, the advantages of using semi-rigid connections in terms of constructional economy and improved control on the structural behavior have been widely recognized. Under seismic excitations, benefits of semi-rigid frames are mainly associated with reduced seismic forces owing to longer structural periods when compared to rigid frames. The seismic performance of semi-rigid steel frames is highly dependent on the cyclic behavior of their beam-to-column connections and joints. In recent years, extensive investigations of the simulation of hysteretic responses and the prediction of low-cycle fatigue endurances of semi-rigid connections and joints under cyclic loading have been conducted experimentally and/ or numerically.

Several types of hysteresis models for the simulation of hysteretic moment-rotation responses of semi-rigid connections have been proposed, including relatively simple analytical linear or non linear models, micro and macro-models based on finite element modeling of connection details. Analytical models and macro-models are more suitable for the simulation of global cyclic behavior of semi-rigid frames, while micro-models are more suitable for the simulation of local cyclic behavior of connections and joints.

Nevertheless, more sophisticated hysteresis models need to be developed and implemented in finite element packages. These hysteresis models should be able to deal with:

 Stiffness-and-strength degradation  Pinching phenomena;

 Asymmetric hysteretic response;  Curvilinear hysteretic response

especially on the unloading path, where piecewise linear hysteresis models may cause remarkable errors in stiffness and/or energy dissipation. Most existing hysteresis models can reproduce hysteresis loops as well as degradation in stiffness, strength and energy dissipation, but they cannot detect failure and consequently cannot assess the ultimate rotation capacity. The failure is generally caused by low-cycle fatigue, owing to the cyclic loading history that causes large inelastic deformation excursions in connections. However, low-cycle fatigue formulae for isolated semi-rigid steel connections under given cyclic loading history cannot be applied directly to predict the low-cycle fatigue endurance in actual frames, since connection

hysteretic responses are unknown and most likely highly dependent on degradation and pinching effect.

The curvilinear cyclic relationship is approximated with a piecewise linear cyclic relationship according to an automatic event definition algorithm.

2.4.2 Formulation of a hysteretic connection element

Connections subject to cyclic reversal loading exhibit gradual degradation in strength and stiffness as well as the consequent degradation in energy dissipation as loading progresses. Pinching effect of hysteretic responses is characteristic of connections that exhibit softening responses under loading reversals for deformations that do not exceed the maxima previously imposed. Failure is unavoidable if both the deformation amplitude and the number of reversals are high enough.

The simulation of cyclic behavior of a hysteretic connection element should reproduce these characteristics. The formulation of a hysteretic connection element comprises the following three major steps:

 First, a multi-linear cyclic skeleton is introduced as an auxiliary piecewise linear constitutive relationship.

 Second, a curvilinear cyclic relationship is reproduced through superposing the linear and non-linear constitutive relationships that approach the linear and elastic-perfectly plastic bilinear components into which the multi-linear cyclic skeleton is decomposed.  Third, the curvilinear cyclic relationship is made to evolve according to both degradation

schemes and pinching schemes, and meanwhile failure is detected through monitoring a damage index.

The hysteretic connection element may be either a translational or rotational spring connecting two nodes with identical coordinates.

2.4.3 The multi-linear cyclic skeleton

A multi-linear hysteretic skeleton is composed of a series of consecutive multi-linear cyclic skeletons evolving with loading cycles. An example of multi-linear cyclic skeleton together with the corresponding experimental data and simulated curve in a typical loading cycle is illustrated in the image below.

Fig. 2.10: example of multi-linear cyclic skeleton [6]

transition from one segment to the next, and a non-linear unloading path is represented by a linear segment. From the multi-linear cyclic skeleton, most of the characteristic quantities defining the curvilinear cyclic relationship such as pinching stiffness, pinching deformation range, loading stiffness, work hardening/softening stiffness, secant unloading stiffness, yield strength, etc. can be evaluated. Based on experimental data, the multi-linear cyclic skeleton is determined according to the following criteria:

 Preserving all behavioral characteristics of the non-linear moment-rotation relationship.  Best fitting the non-linear moment-rotation relationship with the minimum number of

segments. Based on the cyclic skeleton, the corresponding curvilinear cyclic relationship is reproduced.

Based on the evolution of cyclic skeletons, pinching and deterioration parameters are identified, and all hysteresis loops are reproduced consequently.

A typical cyclic skeleton consists of two half-cycles based on the sign of moment, namely, the positive portion (half-cycle) and the negative portion (half-cycle). Every cycle are divided into two reversals (positive and negative). Each portion consists of a loading path and an unloading path.

Fig. 2.11: a typical cyclic skeleton [6]

As the multi-linear hysteretic skeleton is continuous, origins of current half-cycles are equivalent to transition points from unloading to loading of previous half-cycle. Consequently:

The initially inequality is imposed to guarantee positive energy dissipation within a half-cycle. The transition rotation and the transition moment are evaluated according to the following laws:

1) Kinematic work hardening law

Where the superscript indicate the opposite sign of 2) Isotropic work hardening law

This five-segment loading path is the general case. Bilinear and trilinear loading path without pinching, four-segment path without pre-pinching are its special cases. The cyclic skeleton is not required to be symmetric about the origin. Also an originally symmetric skeleton under monotonic loading become asymmetric under cyclic loading owning to stiffness-and-strength degradation, if the degradation is evaluated at each loading reversal.

2.4.4 The curvilinear cyclic relationship – the loading path within half-cycle (j) and loading in

direction (j)

The five-segment linear loading skeleton is decomposed into five components acting in parallel: one elastic with stiffness and four elastic perfectly plastic with stiffness and yield rotation . The relationship can be expressed as:

With: Where:

2.4.5 The curvilinear cyclic relationship – the unloading path within half-cycle (j) and reversing in

direction ( )

The two-segment linear loading skeleton is decomposed into two components acting in parallel: one elastic with stiffness _{and one non linear component with secant unloading} stiffness and tangent unloading stiffness . The relationship can be expressed as: With: Where:

2.4.6 Reference skeleton

A multi-linear reference skeleton is necessary for the evaluation of degradation and pinching. A reference skeleton can refers to the moment-rotation curve under monotonic loading or under cyclic loading. As damage increase owning to cyclic loading, stiffness and strength deteriorate and pinching develops. Hence, the moment-rotation skeleton shifts and shrinks and the energy dissipation decrease. A general reference skeleton relevant to the cyclic skeleton is illustrated in the Fig. 2.12, where the superscript R designate reference.

Fig. 2.12: a general multi-linear reference skeleton [6]

In a conventional bilinear and tri-linear reference skeleton without pre-pinching and pinching, the reference pinching onset moment and pinching rotation range are input as zero, and the reference pre-pinching and pinching stiffness are input as loading stiffness.

2.4.7 Damage indicators, residual functions and pinching functions

Degradation and pinching are evolutionary processes closely related to accumulated damage that can be measured by damage indicator. Two types of damage indicator are introduced into the model of the hysteretic connection element.

 The damage index, denoted is a measure of the damage degree and should be defined according to damage accumulation theories. corresponds to the undamaged virgin stat, while corresponds to failure.

 The damage state variables, convenient normalized to unity, may be considered as monotonically increasing function of damage. Degradation and pinching are expressed as functions of damage state variables. Residual functions are defined the complement of the corresponding degradation function to unity. Since it’s often difficult to define a unique damage state variable that reflects both degradation and pinching, in the follow, the damage state variables of degradation and residual function is denoted whilst the damage state variable of pinching function is denoted . As the result, the hysteretic

response will be better simulated on the basis of better fitting of degraded stiffness-and-strength versus and better fitting of increase pinching versus , respectively.

Either and may be defined using the following cumulative damage indicator formulas:

Where and are respectively the counter of half-cycles and cycles.

may be defined using the above indicated formulas for and and the following cumulative damage indicator formulas:

All the damage indicators are evaluated at previous loading reversal or cycle for two reason: a) Within current loading reversal or cycle, damage is dependent on the previous cyclic

loading history rather than on the current deformation range/amplitude that has not been experienced.

b) No damage exists at the virgin state for the first reversal or cycle.

Stiffness-and-strength residual function can be evaluated using the following formulas:

2.5 Modified Ibarra-Medina-Krawinkler deterioration model with bilinear

hysteretic response

2.5.1 Abstract

Ibarra-Medina-Krawinkler model [8] [9] [10], is a energy-based phenomenological deterioration model used to assess the behavior of steel structures under earthquake loading. The main reason is its ability to capture component deterioration in strength and stiffness. The deterioration model, referred to as Ibarra-Krawinkler model (IK), has been developed by Ibarra in 2005 [8], and then modified by Lignos and Krawinkler in 2009 [10]. A large set of experimental data have been necessary to validate the model, in order to define the parameters that affect the cyclic moment-rotation relationship at plastic hinge regions in beams and to relate them with geometrical and material properties.

Based on information deduced from the steel database, empirical relationship for modeling of pre-capping plastic rotation, post-capping rotation and cyclic deterioration for beams with or without reduced beam section (RBS) have been proposed, as well as information for modeling of the effective yield strength, post-yield strength ratio, residual strength and ductile tearing of steel components subjected to cyclic loading.

2.5.2 Deterioration model

The modified IK model establishes strength bounds based on a monotonic backbone curve (Fig. 2.13a) and a set of rules that define the characteristic of hysteretic behavior between the bounds (Fig. 2.13b). For a bilinear hysteretic response three modes of cyclic deterioration have been defined:

 Basic strength deterioration.

 Post-capping strength deterioration.

 Unloading/reloading stiffness deterioration.

Fig. 2.13 : modified Ibarra – Krawinkler (IK) deterioration model; (a) monotonic curve; (b) basic modes of cyclic deterioration and associated definitions [7]

The model curve is described in terms of moment-rotation quantities. The backbone curve is defined by three strength parameters and four deformation parameters:

      

In addiction the following definition are necessary for an overall comprehension of the model:  Initial stiffness – It defines the elastic branch of the curve.

 Hardening stiffness – It’s defined by connecting the yield point to the peak point. It’s defined as a fraction of the initial stiffness

 Post-capping stiffness - It’s defined by connecting the peak point to the beginning of the residual strength branch. It’s defined as a fraction of the initial stiffness  Residual strength branch – the residual strength is a fraction of the initial yield strength

The rates of the cyclic deterioration are controlled by the Rahnama and Krawinkler rule, based on the hysteretic energy dissipated when the component is subjected to cyclic loading. It’s assumed that every component has a reference hysteretic energy dissipation capacity , expressed as:

Where is a reference cumulative rotation capacity.

Cyclic strength deterioration (basic and post-capping) is modeled by translating the two strength bounds (the lines intersecting at the capping point) towards the origin at the rate

after any excursion in which energy is dissipated. The moment is any references strength value on each strength bound line, and is an energy based deterioration parameter given by

Where

 is the hysteretic energy dissipated at the excursion

 is the total energy dissipated in past excursions (both positive and negative)  is a reference energy dissipation capacity from Eq. 1, and

 is an empirical parameter, usually taken as 1,0.

The same concept apply to modeling of unloading stiffness deterioration, i.e. the deteriorated stiffness after excursion is given by

Parameters of the IK model have been determined by matching the digitized moment-rotation response to a hysteretic response controlled by the backbone curve shown in Fig. 2.14. An example of a satisfactory calibration is shown in Fig. 2.14. A combination of engineering mechanics concept and visual observations is employed to select appropriate parameters and pass judgment on satisfactory matching.

The modified IK model has been implemented in DRAIN-2DX (Prakash et al. 1993) and Open System for Earthquake Engineering Simulation (OpenSees 2010) analysis software.

Collapse prediction of steel moment frame has been validated through comparisons with recent small and full scale shaking table collapse test (Lignos and Krawinkler 2009; Lignos et al. 2010).

Fig. 2.14: Calibration examples of modified IK deterioration model - beam with RBS (no slab, data from Uang et al. 2000) [7]

2.5.3 A new database for deterioration modeling of steel component

The missing aspect of comprehensive modeling of deterioration characteristics of structural components is the availability of relationship that associate deterioration parameters with geometric and material properties and detailing criteria that control deterioration in actual structural element. For this purpose, based on experiment results, three databases have been developed:

 Wide flange beams.  Steel tubular section.  Concrete beams.

The focus is in the first one. The steel database include W-section (mostly beams but also columns). The database contains data in the following three categories:

 Metadata [includes (a) distinction based on configuration of beam-to-column subassembly; (b) connection type; (c) measured material properties of beam and column components; (d) slab details, (e) qualitative summary of the individual test].  Reporter results (measurement and observations, including digitized hysterical load

displacement response, moment-rotation response and panel zone shear force-distortion response).

The steel W (and H) section database documents experimental data from tests have been conducted on beam-to-column subassemblies in which inelastic deformations are primarily concentrated in flexural plastic hinge regions. The primary deterioration mode of the steel components that develop a plastic hinge is local and/or lateral torsional buckling. In the evaluation of modeling parameters presented in the subsequent sections, the data have been subdivided into RBS and other-than-RBS. The latter contain tests of various beam-to-column connection in which a plastic hinge in the beam developed at or near the column face and the pertinent model parameters could be quantified with confidence.

2.5.4 Trends for deterioration modeling parameters

The dependence of modeling parameters ( ) on selected geometric properties of steel section are illustrated by plotting data points of a single model parameter against a pertinent geometric parameter. The information are obtained from calibrations between the IK model and the experimental moment-rotation relationship of the database. A regression line is included in the individual plots to illustrated the overall trends for the modeling parameter. The development of multivariate regression equations that account for correlations of geometric and material parameters in the quantification of modeling parameters are discussing under four trends of data set:

 DS 1: Beams with other-than-RBS connection and depth  DS 2: Beams with RBS connection and depth

 DS 3: Beams with other-than-RBS connection and depth  DS 4: Beams with RBS connection and depth

2.5.5 Statistical information on parameters

Cumulative distribution functions (CDFs) for , obtained from the four data set, are shown in Fig. 2.15 for other-than-RBS and RBS sections. The CDFs reveal general statistical characteristics but do not display dependencies on individual properties.

Fig. 2.15 :Cumulative distribution functions (CDFs) for (a) θp; (b) θpc; (c) Λ. Left: full data sets 1 and 2. Right: data sets 3 and 4, d ≥ 533mm [7]

2.5.6 Dependence of modeling parameter on beam depth

An increase in beam depth is associated with a clear decrease in modeling parameters. Fig. 2.16 shows data and a linear regression line for the pre-capping plastic rotation . One can observe that the trend changes considering only the range of bigger section .

Fig. 2.16: Dependence of plastic rotation θp on beam depth d for beams other-than-RBS (a) full set of data; (b) d ≥ 533mm [7]

2.5.7 Dependence of modeling parameters on shear span to depth ratio L/d

Based on simple curvature analysis with disregards of local instabilities, θp of a given beam section is perceived to be linearly proportional to the beam shear span (distance from elastic hinge location to point of inflection). This perception is supported by Fig. 2.17 which shows the dependence of on .

But the strong dependence is not evident when only beams of are considered. This reason is that most deep beams are susceptibilities to a predominance of web buckling and lateral torsion buckling, and both of these susceptibilities increase with a decrease in moment gradient. This phenomenon offsets much of the curvature integration effect for a larger plastic hinge length. One can conclude that for deep beams a description of beam plastic capacity in terms of a ductility ratio θp/235θy is often misleading because θy increases linearly with L (for a given section) but θp does not.

Fig. 2.17: Dependence of plastic rotation θp on shear span to depth ratio L/d for beams other-than-RBS - (a) full data set; (b) d ≥ 533mm [7]

2.5.8 Dependence of modeling parameters on the depth to thickness ratio of the beam web

This geometric parameter is found to be very important for all three modeling parameters, as shown in Fig. 2.18. The reason is that a beam with large h/tw ratio is more susceptible to web local buckling. This triggers flange local buckling and at larger inelastic cycles also triggers lateral torsional buckling.

Fig. 2.18: Dependence of modeling parameters on h/tw ratio of beam web, d ≥ 533mm [7]

2.5.9 Dependence of modeling parameters on

This ratio is associated with sensitivity to lateral torsional buckling. The parameter is the distance from the column face to the nearest lateral brace and is the radius gyration about y-axis. Tests show that θp is not greatly affected by .

2.5.10 Dependence of modeling parameters on width/thickness ratio of the beam flange

Viewed in isolation, a small ratio has a negligible effect on θp. For most of the deeper beams in the database a small implies a narrow wide flange beam with small radius of gyration and a large h/tw ratio. Larger h/tw ratio makes a beam more susceptible to web local buckling, while small makes a beam more susceptible to lateral torsional buckling. On the other hand the data show a clear benefit of a smaller ratio for the parameters θpc and , since these beams don’t develop a large flange local buckle.

2.5.11 Regression equations for

Regression equations are proposed in order to predict deterioration modeling parameters discussed previously. Since tests have shown that web local buckling is coupled with flange local buckling and lateral local buckling, a non linear model is used to evaluate the contribution of each important property identified previously to select the response parameter (RP). The general non linear model used is:

in which are regression coefficients and are the predictor variables. Six parameter are found to primarily affect the deterioration parameters of steel component. The (2.25) equation becomes: in which is the expected yield strength of the flange of the beam in MPa, normalized by 355 MPa, and and are coefficient for unit conversion, both 1,0 if MPa and mm are used. Equations are presented for beams other-than-RBS and beams with RBS but the aim of this paper is concentrated on the firsts. In this case two sets of equations are proposed: one for the entire range of data and the other for the deep beams with .

2.5.12 Pre-capping plastic rotation

 Full data set – other-than-RBS beams

The large values of regression coefficients for , , and confirm trends pointed out previously. Fig. 2.19 shows data points for predictions obtained by previous equation plotted against obtained data from experimental results based on the calibration process.

Fig. 2.19: Predicted versus calibrated values of θp [7]

 Beams with – other-than-RBS beams

Effects of and on are not significant in this case, as discussed previously.

2.5.13 Post-capping plastic rotation

Fig. 2.20 shows data points for predictions obtained by previous equation plotted against obtained data from experimental results based on the calibration process.

 Beams with – other-than-RBS beams

Fig. 2.20: Predicted versus calibrated values of θpc [7]

2.5.14 Reference cumulative plastic rotation

As already said the reference cumulative plastic rotation is a parameter that defines the rate of cyclic deterioration. All modes of cyclic deterioration are assumed to be defined by the same . The exponent of the equation (2.23) is kept equal to 1,0 for the sake of simplicity. The regression equations are:

 Full data set – other-than-RBS beams

The previous equation indicates that the geometric parameter , and become statistically insignificant.

 Beams with – other-than-RBS beams

2.5.15 Effective yield strength

As mentioned, the modified IK deterioration model doesn’t take account for cyclic hardening. But the effect of isotropic hardening is incorporated approximately by increasing the predicted bending strength to an effective value that accounts for isotropic hardening in average. The following table summarize the mean and standard deviation of for both RBS and other-than-RBS sections.

RBS

Other-than-RBS

2.5.16 Post-yield strength ratio

Post-yield hardening, and subsequently , is described by the ratio of the maximum moment on the backbone curve shown in Fig. 2.13 over the effective yield bending strength .

The and ratios define the strain hardening stiffness of the backbone curve above mentioned. The stiffness is important because of its effect on the stability of a structural system[9]. The following table summarize the mean and standard deviation of for both RBS and other-than-RBS sections.

RBS

Other-than-RBS

2.5.17 Residual strength ratio

Low cycle fatigue experimental studies (Krawinkler et al. 1983; Ricles et al. 2004) indicate four ranges of cyclic deterioration:

 A first range of negligible deterioration in which local instabilities have not occurred or are insignificant.

 A second range which involves an almost constant rate of cyclic deterioration due to continuous growth of local buckles.

 A third range in which deterioration proceeds at a very slow rate due to the stabilization in buckle size. This range is associated with the residual strength of a steel component.  A fourth range of very rapid deterioration, which is caused by crack propagation at local

buckles.

Based on the performed tests a residual strength ratio of about 0,4 is suggested.

2.5.18 Ultimate rotation capacity

At very large inelastic rotations cracks may develop in the steel base material close to the apex of the most severe local buckle, and rapid crack propagation will then occur followed by ductile tearing and essentially complete loss of strength.

The modified IK deterioration model captures this failure mode with the ultimate rotation capacity . This rotation depends on the loading history and may be very large for cases in which only a few very large cycles are executed (e.g. near fault loading). The suggested value of are:

 for other-than-RBS beams.  for RBS beams

For monotonic loading is on the order of three times as large as the values above reported for symmetric cyclic loading protocols.

2.5.19 Conclusion

From the available trend plots, cumulative distribution functions on deterioration parameters, and predictive equations the main conclusions are the following:

 The median value of the pre-capping plastic rotation is on the order of 0,02 rad, the median of post-capping rotation capacity is on the order of 0,20 rad, and the median of the reference cumulative rotation capacity is on the order of 1,0 rad.

 The primary contribution to the deterioration parameters is the beam web depth over thickness . Of some importance is the effect of flange width to thickness , the shear span over beam depth ratio .

 For deep section ( ) a description of beam deformation capacity in terms of

ductility capacity ratio is misleading because increases linearly with while does not.

 Deterioration modeling parameter are not very sensitive to beam span (i.e. the length

of the plastic hinge region).

 The effective yield strength used in the modified Ibarra – Krawinkler model, which

accounts in average for cyclic hardening, is about 1,10 times the plastic moment obtained from plastic section modulus times actual material yield strength.

 The post-yield strength ratio is in average 1,10.

 A reasonable estimate of residual strength is 0,4 times the effective yield strength .  Ultimate rotation capacity of steel components that fail in a ductile manner is

strongly dependent on loading history. For components subjected to symmetric cyclic loading histories is in the order of 0,06 rad, but it’s about three times as large when the component is subjected to a near fault loading protocol or to a monotonic loading.

2.6 Modified Ibarra-Medina-Krawinkler Deterioration Model with Peak-Oriented

Hysteretic Response

2.6.1 Abstract

The first modified version of the original Ibarra-Medina-Krawinkler deterioration model have been developed by Ibarra et al. in 2005 and then by Lignos and Krawinkler in 2011 [10] at Stanford University. The advantage of this model is that can be used for collapse assessment of deteriorating structural systems subjected to earthquake loading.

2.6.2 Model

This model keeps the basic hysteretic rules proposed by Clough and Johnson in 1965, but the backbone curve is modified to include strength capping and residual strength. The presence of a negative post-capping stiffness does not modify any basic rules of the model. Fig. 2.21

shows the deterioration of the reloading stiffness for a peak-oriented model once the horizontal axis is reached (points 3 and 7). The reloading path targets the previous maximum displacement.

Fig. 2.21: Peak-orientated hysteretic model – basic model rules [9]

Mahin and Bertero (1975) proposed that the reloading path be directed to the maximum displacement of the last cycle instead of the maximum displacement of all former cycles in the former path results in a larger reloading stiffness. This is exemplified in Fig. 2.22, where the reloading from point 10 is directed to point 7 (maximum displacement of last cycle) instead of point 2 (maximum displacement of all earlier cycles), as in original peak-orientated model. Once point 7 is reached the path is redirected to point 2.

2.7 Modified Ibarra-Medina-Krawinkler Deterioration Model with Pinched

Hysteretic Response (ModIMKPinchingMaterial)

2.7.1 Abstract

The second modified version of the original Ibarra-Medina-Krawinkler deterioration model have been developed by Ibarra et al. in 2005 and then by Lignos and Krawinkler in 2011 [10] at Stanford University, parallel to the first. Also in this case the advantage of the model is that can be used for collapse assessment of deteriorating structural systems subjected to earthquake loading. It also include pinching deterioration in the formulation.

2.7.2 Model

The Pinched model is similar to the previously defined peak-oriented one, except that reloading consists of two parts. Initially the reloading path is directed towards a point denoted as “break point”, which is a function of the maximum permanent deformation and the maximum load experienced in the direction of loading. The break point is defined by the parameter , which modified the maximum “pinched” strength (point 4 and 8 of Fig. 2.23), and , which defines the displacement of the break point (points 4’ and 8’). The first part of the reloading branch is defined by and once the break point is reached (points 4’ and 8’), the reloading path is directed towards the maximum deformation of earlier cycles in the direction of loading by .