Dipartimento di Analisi e Progettazione Strutturale, Facolta diIngegneria, Universita di Napoli Federico II, 80125 Napoli, ItalyE-Mail: gamanfre @ unina. it
Abstract
Most of the existing reinforced concrete buildings were designed according toearly seismic provisions or, sometimes, without applying any seismic provision.Some problems of strength and ductility, like insufficient shear strength, pull-outof the rebars, local mechanism, etc., could characterize their structural behaviour.Therefore the presence of these problems requires a refined procedure ofassessment of existing r.c. structures. For this aim an useful tool can be thepushover analysis of the structures, that is suggested by different advanced codeand provisions.In this paper an innovative numerical model is presented. This model allows totake into account the most important mechanical phenomena that influence thenon-linear behaviour of the reinforced concrete frames. The push-over analysis ismade using a refined point by point model including an explicit formulation ofthe bond slip relationship and capable to take into account the effect of thedistributed and concentrated non-linearity as the spread of plasticity along themember and the fixed end rotation. The results of the pushover analysis of anexisting underdesigned building are also presented. This analysis includes alsothe estimation of the local demand of curvature ductility in beams and columnsthat is related with the interstorey drift.
1 The global behaviour of existing r.c. frames
Reinforced concrete frames designed according to early seismic provisions or,sometimes, without applying any seismic provision, usually have a low strength
and in most cases they show a limited amount of ductility and the absence of theappropriate hierarchy of failure modes (Cosenza and Manfredi, [1]). Moreover,the general design of the entire structures could be insufficient, being somedesign requirements (i.e. regularity in plan and in vertical, absence of weakstory) missing. Very often details are poor (i.e. low percentage of stirrup, poorbond conditions). Consequently the critical zones (beam-column joints, footingzone of beam column) do not behaviour in a ductile way, showing a brittlemechanism of failure (i.e. rebars pull-out, buckling of bars, shear failure, etc.).
The above mentioned topics lead to a number of problems in the evaluation ofthe seismic behaviour of r.c. frames. In general, all the deterioration sources arepotentially active, and this occurrence potentially complicates all the steps of theseismic assessment.
In particular for the beams the major problem occur in the ends where,considering seismic actions, a brittle shear failure could occur, because of thesuperposition of shear forces due to vertical loading and seismic loading. Thesituation of joints could be critical, especially for the exterior joints wherelongitudinal re-bars are not continuous, due to low amount of stirrups and morein general to poor details.
In the vertical elements, collapse for concrete crushing could appear inconsequence of insufficient confinement. The problem of column shear failure isvery serious because it could lead to the catastrophic collapse of the structure,due to loss of equilibrium. Due to a small number and poor detailing of stirrups,also problems of local buckling of re-bars in the footing zone and more ingeneral at the ends of columns could appear.
For what concerns the behaviour of elements under flexure with high shear, it isnecessary to introduce refined models capable to estimate shear strength and totake into account the decrease of flexural ductility of members due to theinteraction with shear in order to evidence the presence of brittle shear failure inthe global frame behaviour.
The absence of the hierarchy of failure philosophy has important consequence inthe global behaviour of existing frames. In general poor dissipation capacitymechanism could occur: mixed sides way mechanism with plastic hinges andshear failures could appear. In particular main problems are:
- shear failure in the beams could appear. These types of failure are brittle andlead to a small amount of global ductility.
- the design of columns for vertical load lead to interior column with large crosssectional area and small reinforcements. The result is the presence of interiorcolumns with high stiffness and low strength. In terms of global mechanism theresults is that plastic hinges at the ends of interior columns could develop.
Summarizing, a number of unexpected mixed mechanisms with brittle failuresand a poor global behaviour could appear.
The problem is substantially different from the case of frames designedaccording to new generation codes, where the aim is to avoid poor mechanisms.As a consequence effective tools to check the global behaviour of the frame arerequired.
The recent provisions (BIA [2], ATC [3]) move in this direction. In fact a non-linear static push-over analysis is suggested in order to evaluate the allowableglobal ductility or the accepted maximum interstorey drift as measures ofstructural performances of the frames. The interstorey drift, that is defined as therelative lateral displacement between two adjacent floors divided by the storeyheight, can be related with the intended use of the structure and the propertiesand performance characteristics of non-structural elements.
The global ductility is defined as the maximum top-storey displacement of thestructures divided by the global yield displacement of the structure. The globalyield displacement is evaluated also using the non-linear static push-overanalysis of the structures. These global measures of seismic performance of theframes are approximate and incomplete, but they are sufficient for design scope.
The push-over analysis (Moehle [4]) is based on an incremental static analysis ofthe structure using a prescribed lateral load distribution, scaled incrementally byan amplification factor. All gravity loads that are considered in the code-specified load combinations should be applied to the structure before applyingthe lateral loads. The output of this procedure is a relation between the top storeydisplacement (Collins et al.[5]) or the displacement at centre of action of lateralseismic force (BIA [2]) and the amplification factor of the lateral loads. Thisdiagram can be substituted by an equivalent bilinear curve and provides theglobal yield displacement, the allowable global ductility, the required interstoreydrifts and the period of the equivalent system model.
Obviously the reliability of the results depends on the development of a reliablenumerical model of the frame in terms of behaviour mechanisms and materialproperties.
In this paper a refined model and numerical procedure for the non-linear analysisof reinforced concrete frames is introduced. The seismic assessment of anexisting building in Catania is briefly presented in order to show the capabilitiesof this model.
2 Modelling of reinforced concrete frames
In the evaluation of existing frames reliable models to analyze the behaviour ofthe structure under seismic loading are needed. In the following a very briefsummary of models that permits the analysis of the non-linear behaviour of r.c.structures is described. These can be divided into three categories in accordancewith the increasing level of refinement and complexity:
Global models. The non-linear response of a structure is concentrated at selecteddegrees of freedom. For example, the response of a multi-storey building can berepresented as a system with one lateral degree of freedom for each floor. Eachdegree of freedom has the hysteretic characteristics of the interstorey shear-lateral drifts response. Such models are useful in the preliminary design phasefor estimating interstorey drifts and displacement ductility demand. Thereliability of this class of model in the accurate prediction of global displacement
is poor and the recovery of internal member forces from the limited number ofdegrees of freedom is practically impossible.Member by member models. The structure is modelled as an assembly ofinterconnected elements that describe the hysteretic behaviour of reinforcedconcrete members. Constitutive nonlinearity is either introduced at the elementlevel in an average sense or at section level. Correspondingly, two types ofelement formulation are possible: lumped nonlinearity and distributednonlinearity member models (Takizawa [6], Arzoumanidis et al. [7], Soleimaniet al. [8], Filippou and Issa [9], Otani [10], Kunnath et al. [11]).Point by point models. Members and joints are discretized into a large number offinite elements Constitutive and geometric nonlinearity is typically described atthe stress-strain level or averaged over a finite region. Bond deteriorationbetween steel and concrete, interface friction at the cracks, geometric crackdiscontinuities are among the physical nonlinearities that can be studied with thisclass of models (Kaba and Mahin [12], Zeris and Mahin [13], Taucer et al.[14],Spaconeetal. [15]).
3 Current capabilities of the proposed model
The current version of the model proposed in the present paper is brieflydescribed. The beam/column element is a fiber model with cracking and spreadplasticity considered as a point-by-point element. The model takes into accountthe mechanical and geometrical nonlinearity, including P-A effects.The mechanical characteristics of the cross section are evaluated from thematerials properties, introducing appropriate constitutive relationships.Particularly, for the columns the moment curvature relation varying the axialload level on the section is evaluated.The beam-column are divided in a number of elements in relation to thegeometrical and mechanical changes; it is possible, for example, to take intoaccount the variation of reinforcement between the end zones and the middlezone of the beams.The structural model overcomes the hypothesis of no slip between steel rebarsand concrete including the bond stress-slip relationship. This last characteristicallows a detailed evaluation of tension stiffening effect both in elastic and post-yielding field. For this purpose, each beam element is divided in sub-elementsdefined by two consecutive cracks. The cracks should occur in the sectionswhere stress in concrete reaches the tensile ultimate value; otherwise, the
distance between cracks Al can be calculated using the semi-empiricformulations provided by codes. Therefore the proposed element assumes thatcracks position is known since the beginning of the loading, but the cracks openwhen the bending moment is greater than the cracking one in the section.The analysis of the sections between two successive cracks is made byconsidering that concrete in compression and steel in tension are strainedaccording to the Bernoulli hypothesis: the plane cross-sections remain plane.The sub-element among two cracks is solved introducing the steel-concrete bond
between bars and concrete, so that the tensile stress of steel is transferred to aneffective area around the bar of concrete in tension, where the concrete stress isassumed constant. Introducing the constitutive bond-slip relationship i-s, thenumerical solution of the bar equilibrium and of its congruence with thesurrounding concrete is developed by a finite difference method. Through thisprocedure, it is possible to evaluate the tension stiffening effect in terms ofaverage curvature of the sub-element.The integration of the average flexibility of the element, along the length of theelement, provides the matrix of flexibility of the element and subsequently thematrix of rigidity (Manfredi and Pecce [16]).The aforesaid approach allows to avoid the approximations due to the definitionsof the plastic hinges zone. For the column elements results possible to take intoaccount the variation of axial load caused from the presence of horizontal loadsand their influence in terms of strength and deformation of the section.The fixed end rotations in the joints and in the footing zones are evaluated by thesame approach previously described; for a given anchorage length of the rebarsthe spreading of the yielding is evaluated, together with the consequent slip atthe interface between elements and nodal zones.For what concerns the shear effects in the post yielding field phase, the modeltakes into account the reduction of the shear capacity due to the flexural ductility,introducing the relation proposed by Priestley et al. [17].In conclusion the model is capable to evaluate the deformation and the strengthcapacity of the frame in the non linear range, considering both a globalmechanism and/or a local collapse due to bending and/or shear.
4 A case study of an existing building in Catania
The proposed model is tested in the non-linear analysis of an existing building,designed only for gravity load in the Seventies. The building has a rectangularplant with a symmetrical axis in the longitudinal directions and has three levels.A pushover analysis of a plane frame in the transversal direction is presented.The geometrical dimensions of the elements, the reinforcement percentage, thestructural masses and the loads are obtained from the original draws. Thestrength of concrete and reinforcing steel are assumed equal to the declaredvalues. For the concrete in compression a conventional parabola-rectangularrelation is introduced while for the steel the relation proposed by (Shima et al.,[18]) is used assuming an ultimate elongation typical of the steel used in theSeventies in Italy. The main materials characteristics are summarized in table 1.The moment-curvature relation for each section is obtained by means of a stripmethod introducing the material characteristics and the axial load. The tension-stiffening effect is obtained step by step introducing the bond properties with theBertero-Eligehausen-Popov [19] relationship. For the footing zone of thecolumns and for the node regions of the beams the fixed end rotation is takeninto account by means of a moment rotation relationship, evaluated starting fromthe calculation of the slip at the joint interface due to the yielding penetration and
Strain at yielding, £yStrain at strain-hardening, &*Strain at ultimate stress, 8u
380.00 (N/nW)475.00 (N/mnf)
0.18%20%14%
Table 1. Material properties used in the case study
The plane frame is subjected to horizontal loads with a triangular distributionalong the height; the gravitational loads are concentrated on the upper side of thecolumns.The pushover curve, obtained with the proposed model, is drawn in Figure 1. Inthe graph the total displacement of the upper level of the frame is drawn on thehorizontal axis versus the base shear on the vertical axis. The analysis of thecurve shows the little first linear branch and, after the first cracking, a non-linearbehaviour due to the cracks propagation in the elements.
250
200 -
150-
100 -
50-
0.35p 0.35p 0.25p 0.28pPUSH-OVER CURVE
• NOMINAL YIELDINGA LOCAL COLLAPSE+ FIRST ELEMENT YIELDING
The first yielding becomes for a base shear level about equal to 116 kN and it isrepresented in the figure 1 by a solid square. This first yielding not significantlychanges the stiffness of the frame that is sharply modified for a higher level ofthe base shear corresponding to the incoming mechanism development. Howeverthe ultimate value of the curve corresponds to a local crushing in a base column(8=7.95 cm). In the figure a bilinear schematization of the pushover curve is alsodrawn. The conventional yielding point is obtained assuming a displacement(8=2.45 cm) correspondent to the development of the 50% of the plastic hingesin the collapse mechanism of the structure (SEAOC [20]). The conventionalvalue of the yielding base shear (F=169kN) is obtained using the energyequivalence criterion between the actual pushover curve and the bilinearschematization.On the same curve the displacements relative to the intermediate structuralperformance levels are drawn according to the suggestions reported in [20]. Theglobal ductility of the analized frame results equal to 3.19 and the collapse isachieved for the concrete crushing in a column of the first level. In the followingthe local damage in the elements of the first level is discussed. In figure 2 thebase shear coefficient Q>, equal to the ratio between the base shear and thegravitational masses, is drawn versus the interstorey rotation AS/Ah of the firstfloor. The global structural performance levels, evaluated as percentage of thetotal plastic displacement, are also drawn.
0.14
0.002 0.004 0.006
A5/Ah
0.008 0.01
Figure 2. Development of the plastic curvature in the elements
In the same figure 2 the ratio between ultimate curvature and yielding curvature,assumed as measure of the local damage, is also drawn versus the interstoreyrotation A8/Ah. The comparison between global performance levels and localplastic damage levels suggests the following remarks:
• the first yielding in the elements occurs before the achievement of the firstperformance level SP-1 that correspond to the conventional yielding; thefigure 2, where the development of the plastic curvature in the elements ofthe first floor is drawn, shows that about the 50% of the elements is yieldedat the level SP-1;
• the columns yield after the plasticization of all the beams and it provides asharp increase of the plastic curvature demand. This behaviour is due to thehigh level of axial load that provides for the columns a reduction of theelement ductility;
• the frame collapse (performance level SP-5) is achieved for a value of Cyequal to about 0.12, corresponding to the concrete crushing of a basecolumn.
The model allows to obtain a micromodelling of joints: in figure 3 the steelstrains, the slip and the bond stresses distributions along the bar in a joint of theexamined frame are drawn for a level of stress in the rebars equal to fu.The steel strain distribution shows the spreading of yielding in the core of thejoint and the consequent slip of the rebars that determines the fixed end rotationat the interface joint-beam. The bond strength distribution shows a debondinglength near the joint interface due to the degradation of bond-slip constitutiverelationship included in the model.
5 Conclusions
The current version of the model proposed in the present paper seems capable todescribe the non-linear behaviour of underdesigned reinforced concrete framesincluding brittle modes of failure. Detailed description of bond-slip problems isprovided The analized frame, representative of the existing structures built inCatania in Seventies, shows a low strength and ductility. The collapse isachieved for concrete crushing in a column of the first floor.
Acknowledgements
The research presented in this paper was partially supported by the ItalianNational Group for the Defence against Earthquakes (GNDT) in the ResearchProject "Catania".
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