nonlinear models and nonlinear procedures for ...

ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis, D.C. Charmpis, N.D. Lagaros, Y. Tsompanakis (eds.) Rethymno, Crete, Greece, 13–16 June 2007

NONLINEAR MODELS AND NONLINEAR PROCEDURES FOR SEISMIC ANALYSIS OF REINFORCED CONCRETE FRAME STRUCTURES E. Spacone1, G. Camata2, and M. Faggella3 1

University of Chieti-Pescara Dept. PRICOS viale Pindaro 42, 65127, Pescara, ITALY e-mail: [email protected] 2

University of Chieti-Pescara Dept. PRICOS viale Pindaro 42, 65127, Pescara, ITALY e-mail: [email protected] 3

University of Chieti-Pescara Dept. PRICOS viale Pindaro 42, 65127, Pescara, ITALY also, Department of Structural Engineering University of California San Diego 9500 Gilman Drive, MC 0085 La Jolla, California 92093-0085 e-mail: [email protected]

Keywords: Nonlinear frame analysis, nonlinear models, pushover, fiber section, plastic hinges, reinforced concrete. Abstract. Nonlinear methods of analysis for seismic vulnerability assessment of existing structures are attracting increasing attention for their capacity to predict the actual seismic behavior better than linear methods. Their inclusion in modern design codes such as Eurocode 8 is a major boost for their use in practical applications. Recent years have also seen steady advances in the modeling capabilities of reinforced concrete frames, particularly in the field of beam and column behavior. Advanced frame modeling tools are included in both research and commercial codes. As research and practicing engineers start applying nonlinear methods to real problems, questions arise regarding the application of the design codeprovided procedures and the selection of the appropriate structural model given the objective limitations of the frame modeling capabilities today available. The scope of this paper is to discuss some of the issues still open regarding the applications of the nonlinear methods of analysis to reinforced concrete structures, with the firm belief that an open discussion on these methods, their advantages and limitations will open the way to a more widespread use of advanced and more accurate methods of analysis.

E. Spacone, G. Camata and M. Faggella

1

INTRODUCTION

Modern seismic design codes allow engineers to use nonlinear methods of analysis to compute design forces and design displacements. In particular, two nonlinear methods are considered: simplified nonlinear pushover analysis and full nonlinear time-history analysis. These methods refer to the design and analysis of frame structures, mainly buildings and bridges. Nonlinear methods are of particular interest for the seismic vulnerability assessment of existing structures, where the modern concepts of capacity design have not been used, and therefore the application of linear methods of analyses cannot be easily justified. The is no doubt that nonlinear analyses are more accurate than linear analyses in reproducing the actual structural behavior up to the Ultimate and Collapse Limit States, but there still lacks the necessary experience to make nonlinear methods of analyses routine methods for existing structures. The two nonlinear methods require advanced models and advanced nonlinear procedures in order to be fully applicable by design engineers. Furthermore, reinforced concrete (RC) structures show a highly nonlinear behavior that is not easily modeled with frame elements. Concrete and steel nonlinear response, concrete cracking, bond-slip, rebar debonding, rebar buckling, shear failure, joint deformations, joint failure, progressive damage are some of the main sources of nonlinearities during a moderate or strong seismic event. Due to the relatively recent introduction of the nonlinear methods, the nonlinear procedures are not yet fully applicable and not many nonlinear programs are available to run them. This paper intends to summarize the two nonlinear procedures contained in Eurocode 8 [3] (EC8), to highlight some of the open issues related to their applications, and to discuss them with the aid of two applications, one to a RC building, the other to a RC bridge. 2

OVERVIEW OF NONLINEAR METHODS OF ANALYSIS

This brief overview deals mainly with the nonlinear procedures in EC8. Other seismic codes contain similar approaches ([1],[11]). Two nonlinear methods of analysis are considered in EC8: the nonlinear pushover analysis (NPO) and the nonlinear time history analysis (NTH). The initial steps of both nonlinear procedures are identical: construction of the nonlinear frame model and application of the gravity loads. The gravity loads remain constant during the nonlinear analysis. This initial step is quite important because it changes the initial state of the structure. In a RC building the gravity loads typically induce cracking in the beams and apply high axial forces to the columns. 2.1

Nonlinear Pushover Analysis

The NPO consists of applying monotonically increasing constant-shape lateral load distributions to the structure under consideration. The model can be planar or three-dimensional, depending on the plan irregularity of the structure. For buildings conforming with the regularity criteria of EC8 the analysis NPO may be performed using two planar models, one for each principal horizontal direction. For plan-irregular buildings, two independent analyses with lateral loads applied in one direction only are performed, and the results are then combined. Plan irregularity is typical of older structures. The NPO procedure in EC8 follows the N2 method developed by Prof. Fajfar [9]. The N2 method was based on rigid frames, where the floor beams are rigid. The N2 method consists of applying two load distributions P = MΦ Φ (where M is the mass matrix and Φ the load shape array):

2


a) a “modal” (or “triangular”) pattern with Φ = Φ1 , that is P1 = MΦ Φ1 , where Φ1 is the first mode shape in the load direction considered; b) a “uniform” or “mass proportional” pattern with Φ = R , that is P 2 = MR , where R is the influence vector. In the N2 method Φ1 is normalized so that the top floor (n) displacement is 1, i.e. Φ1,n = 1 . The two load distributions are schematically shown in Figure 1. The applied lateral load distributions are increased and the response is plotted in terms of base shear Vb vs. top floor displacement D (for example center of mass of the top floor). This is the so-called pushover curve or capacity curve (also shown schematically in Figure 1). The above load patterns are supposed to reproduce static loads equivalent to the seismic action. The two load patterns correspond to a first-mode dominated behavior and to a bottom soft-story response. Other distributions can be used but are typically non considered. Pi2=miΦ1,i Pi1=mi

D1

Vb 1 Base Shear

D2

2

2

1

Top Displacement

Base Shear Vb

D

Figure 1 Load distribution for pushover analysis according to EC8 and pushover response curves

The N2 procedure transforms the response of the MDOF system into the response of an equivalent SDOF system, with force-displacement curve F * − D* , with F * = Vb Γ , D* = D Γ

and Γ = ( Φ T MR ) ( Φ T MΦ ) . This is necessary in order to compare the building capacity

T*

SA /g

TC

curve of Figure 1 with the demand, expressed in the design codes by the design spectra, which refer to SDOF systems ([8],[9]).

Medium and long period T* µ=1 (elastic)

µ

D*y Dt* = Det*

µ=

Det* D *y

SD (m)

Figure 2 Capacity and Demand spectra for medium and long period T*: determination of the target displacement for SDOF

In order to compare elastic design spectrum (demand) and capacity curve, first the capacity curve is transformed into an energy equivalent elastic-perfectly plastic system, then the ADRS (Acceleration Displacement Response Spectrum) representation is used, as shown in Figure 2.

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Approximate nonlinear demand spectra are obtained using simplified formulas [3]. The design point is given by the intersection between the capacity curve and the inelastic response spectrum of equal ductility. Once the target displacement Dt* for the SDOF system is determined, the target displacement for the MDOF is computed as Dt = ΓDt* . All the design verifications for the given Limit State are performed at the top floor displacement D = Dt . For non symmetric buildings, two different pushover analysis should be performed for each seismic input direction, in the positive and negative directions. For structures that are torsionally flexible, EC8 suggests to account for the torsional effects by following a simplified procedure where the frame displacements are increases based on a modal response spectrum analysis. This approach is based on the modified N2 procedure [10]. The pushover procedure described so far refers to the application of the ground motion in a single direction. When the ground motion is applied in the two horizontal directions x and y, (this is the case of plan irregular buildings, i.e. the great majority of existing buildings in southern Europe) EC8 gives two alternatives for the combination, the SRSS method or the 100/30 Combination rule (Figure 3). Fby

Py

y Dty

Dy

EEdy

Combination rule 1

SRSS 2 2 EEd = EEdx + E Edy

Fbx

Combination rule 2

Px EEdx

x Dtx

E Ed = E Edy + 0,3E Edx E Ed = E Edx + 0,3E Edy

Dx

Figure 3 EC8 Combination rules for horizontal seismic forces

In nonlinear analysis, the action effects (as they are called in EC8) may be either forces or deformations. For ductile mechanisms (such as flexure in beams) the action effects are the deformations (for flexure, plastic hinge rotations or curvatures), while for brittle mechanisms (such as shear) the action effects are forces (shear forces in the structural elements). EC8 gives some indications on how to check the seismic performance of a building based on nonlinear analysis results, but studies are still under way and a clear checking procedure is still missing The above combination appears as an extension of the directional combination rules in linear procedures. It remains to be proven that they can be extended to the nonlinear case. It seems more logical to apply the two load patterns at the same time, but this topic is still to be studied. The N2 method was originally developed for plane frame. Its extension to space, plan-irregular buildings with mixed translational-torsional modes, modeled with advanced modeling tools (see later) is a challenge.

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Another important issue with the N2 methods stems from the fact that the applied load shapes remain constant during the load history, thus the progressive damage of the structure does not affect the applied seismic loads. One alternative pushover method is the so called adaptive pushover. One example is the method developed by Elnashai, Pinho et al. ([2],[4]). The method adapts the form of the applied load to the inelastic evolution of the structure, given by the instantaneous stiffness matrix of the structure. In other words, at given points along the pushover curve, the applied load shape is updated based on the instantaneous mode shapes. This is typically done using either total or incremental updating. In total updating, the total load vector is changed based on the new mode shapes. In the incremental updating form, only the increment of lateral loads includes the change in stiffness and thus the instantaneous mode shapes. This method is an improvement on the N2 method, but the changes in loading function cannot fully reflect the cyclic damage state the structure can have in the actual dynamic motion. Another method is the so-called modal pushover developed by Chopra and Goel. [6]. The idea is to extend the modal analysis from the elastic to the inelastic range. It is assumed that the response of the system can still be expressed in terms of n uncoupled nonlinear equations of motion. If the excitation is proportional to the i mode, it is assumed that the response is still dominated by the same mode. That is, if the load is proportional to Φi , the displacement response is dominated (and thus proportional) to Φi . This allows modal decomposition. The global response is obtained combining the modal responses. The pushover procedure is repeated for each mode, and the resulting target displacements are found for each mode. The maximum seismic effects are found using the SRSS or the CQC modal combination rules. The procedure is obviously approximate, but the results published so far show improved results when compared with the N2 method of EC8. The main criticism to this method is that it combines responses of totally independent structures (the single bilinear SDOF systems) and thus lacks a sound physical interpretation. Another problem, common to other nonlinear pushover procedures, is the modal combination based on SRSS or CQC, where the sign of the action is lost. An alternative nonlinear static procedure was proposed by Aydinoglu [5]. The procedure, called Incremental Response Spectrum Analysis (IRSA) does not properly belong to the pushover methods. The procedure is displacement-based, it uses the equal displacement rule and the structure nonlinear behavior is modeled as piece-wise linear. The initial modes are computed and the elastic response spectrum for the initial structure is carried out. In the first stop the spectral ordinates are scaled to the formation of the first plastic hinge, which corresponds in the piece-wise linear capacity curve, to the first change of stiffness. The updated modal quantities are computed and additional spectral ordinates are computed up to the formation of the second plastic hinge. The procedure continues until the entire spectral ordinates are applied. The method does not require any transformation to an equivalent SDOF system. Other pushover procedures have been proposed in the published literature, but no general consensus has been reached on the best possible procedure. It appears that at this stage the N2 method represents a good compromise between simplicity and precision. A number of studies are however under way in this field.

2.2

Nonlinear Time History (NTH) Analysis

The nonlinear time-history analysis is the most general nonlinear method of analysis. It is deemed unpractical by some because of the computational length of the analysis, but as computer speeds increase, NTH becomes increasingly feasible. Time-history analysis is a general dynamic method of analysis and there are no computational peculiarities related to earthquake 5


engineering analysis. The input force is the input ground motion, expressed in the form of one or more accelerograms. The dynamic equations of motion are:

ɺɺ + CU ɺ + F( U) = − MRu ɺɺ g MU

(1)

where C is the damping matrix, ɺuɺ g is a vector containing accelerograms and depends on

ɺɺ g ( t ) . In the most general case, u ɺɺ g = {uɺɺg , x time, i.e. u

uɺɺg , y

uɺɺg , z } , that is the input ground

motion, consists of three simultaneously acting accelerograms in three orthogonal directions. For buildings, the z component of the ground motion is typically neglected. Thus, for a 3D structural model, only the horizontal components are considered and ɺuɺ g = {uɺɺg , x uɺɺg , y } . There are still a few unresolved issues related to NTH analysis: the selection of the appropriate structural model (this problem applies to NPO analyses too), the selection of the ground motions, the selection of a correct damping matrix, the heavy computational effort, the large volume of output quantities to be analyzed. As for the selection of the ground motions for the NTH analyses, EC8 states that: a) a minimum of 3 different pairs of accelerograms ɺuɺ g = {uɺɺg , x uɺɺg , y } should be used. If 3 to 6 ground motions are used, then the maximum of the response quantities from all the analyses should be used as the design value of the action effect Ed in the relevant verifications; b) if 7 or more different ground motions are used, i.e. if 7 different pairs of accelerograms ɺuɺ g = {uɺɺg , x uɺɺg , y } are used, then the average of the response quantities from all the analyses should be used as the design value of the action effect Ed in the relevant verifications. Depending on the nature of the application and on the information actually available, the description of the seismic motion may be made by using artificial, recorded or simulated accelerograms [3]. EC8 establishes rules with regard to obtaining spectrum-compatible accelerograms. Though no extensive study is yet available and discussions are under way in those countries that have started employing nonlinear methods of analysis, it is generally recognized that recorded accelerograms, if well selected, are less demanding on the structure, because each single accelerogram does not match well the elastic response spectrum, and the frequency content of each single accelerogram is limited with respect to the elastic response spectrum. On the other hand, generated accelerograms are more demanding, because each single accelerogram has a response spectrum close to the elastic response spectrum. Some claim that generated or simulated accelerograms should be used at the research level for the refinement of the nonlinear methods of analysis, while recorded accelerograms should be used for the actual design of structures. No general consensus has yet been reached, though. A few computer programs are freely available on the internet for generated and simulated accelerograms. As for recorded accelerograms, there are databases available (such as the European Strong Motion Database (http://www.isesd.cv.ic.ac.uk), the PEER Strong Motion Database (http://peer.berkeley.edu/smcat/index.html ), the Strong Motion Database of SeismoLinks (http://www.seismolinks.com/Strong_Motion_Databases/) to mention just a few. The elaboration of these databases of recorded ground motions and the selection of the appropriate accelerograms that satisfy the EC8 requirements is not trivial. An effort to extract and filter the accelerograms of the European Strong Motion Database is underway in the Italian project Reluis [13]. Groups of accelerograms compatible (according to EC8 criteria) with the elastic re-

6


sponse spectra for different ag and soil type are available at http://www.reluis.unina.it/ (in Italian). EC8 does not contain any specific procedure for the estimation of the torsional effects. In principle, the general procedure for accidental torsional effects should be applied. However, this would imply applying the ground motions at 4 different locations, with a considerable increase in computational costs. At this stage it appears reasonable to neglect the torsional effects when performing nonlinear dynamic analyses. Another important issue related to NTH analyses is the selection of the damping matrix C in Eq. (1). The commonly used rule of employing Rayleigh damping with 5% damping in the principal modes cannot be applied here. First of all, the stiffness matrix changes at every instant. Some programs allow for updating of the Rayleigh damping C matrix as K changes. But the biggest issue concerns which modal damping ratio values to use. Several analyses have shown that 5% is too high a damping for an inelastic structure, as is easily understood ɺ accounts already for some nonlinearity in the structure. If the masince viscous damping CU terial nonlinearities are accounted for in the nonlinear static term F( U ) , then the damping ratio should be reduced. The more accurate the nonlinear structural model is, the lower the damping ratios should be [20]. It appears that a damping coefficient between 2% and 3% is suggested by most analysts. The issue is however very important (and difficult to solve) because for very low damping, small changes in damping imply large variations in the results. NTH analyses typically produce large amounts of output data. This problem is related to the fact that there is no consensus yet on the verification procedures. Once the verification procedures are well established, it will be a lot easier for commercial and research software to contain automatic design checks.

3

OVERVIEW OF CURRENT NONLINEAR MODELS

There are two main sources of nonlinearities; material and geometric. Material nonlinearities have received more attention in earthquake engineering analyses, but geometric nonlinearities may be quite important when dealing with the ultimate and collapse limit states. EC8 does not have any specific guidelines on how to model material nonlinearities, except for general considerations on the concrete and steel constitutive laws ([3]). Material properties should be based on mean values of the mechanic characteristics for existing structures. EC8 provides some guidelines on geometric nonlinearities, but only on when and how to account for them in an approximate way. Second-order, or P-∆ effects, need not be taken into account if the following condition is satisfied:

θ=

Ptot d r ≤ 0,10 Vtot h

(2)

where θ is the interstorey drift sensitivity coefficient, Ptot is the total gravity load at and above the storey considered in the seismic design situation, d r is the design interstorey drift, , Vtot is the total seismic storey shear, h is the interstorey height. If 0,1 < θ ≤ 0, 2 , then nonlinear geometric effects may approximately be taken into account by multiplying the relevant seismic action effects by a factor equal to 1 (1− θ ) , θ > 0,30 is not accepted. This is obviously an approximate, code-oriented approach to geometric nonlinearities, while inclusions of the second-order effects typically depends on the capabilities of the nonlinear program used. Material nonlinearities are typically modeled with either lumped or spread plasticity models. Lumped plasticity models model beams and columns as linear elastic, except in given

7


points where plastic hinges can form. Plastic hinges can be specified in many different forms, using ad hoc plastic laws or more advanced fiber-based section models. In order to model the plastic hinge behavior, it is necessary to assess the length of the plastic hinge Lpl, though no general formula or procedure exists. The plastic hinge length is important to transform plastic rotations into plastic curvatures, and vice-versa. In some programs it is also possible to include plastic hinges that describe shear failure (plastic hinge is not probably the right label in this case!). This is important for shear deficient structures. Most commercial software includes lumped plasticity models. For inexperienced analysts, plastic hinges are probably easier to handle than spread plasticity models. The main shortcoming is that structures are assumed mostly elastic, thus approximations are needed for capturing the member elastic stiffness. Spread-plasticity models are classical finite elements where material nonlinearities are modeled at each integration point. Besides the classical two-node displacement-based beam elements, force-based two-node force-based elements have seen a widespread use both in research and commercial software ([15],[17],[25]). The assumed force fields in a two-mode force-based element are exact within classical beam theories, such as the Euler-Bernoulli and Timoshenko theories ([14],[24]). This implies that only one element per structural member is used. The element implementation is not trivial and it implies element iterations, but these steps are transparent to the user. The section constitutive law is the source of material nonlinearities. In principle, the same section constitutive laws can be used for either spread-plasticity elements (at the integration point level) o in lumped models (at the plastic hinges). Fiber section models are considered state-of-the-art at present, because they automatically account for the interaction between axial and bending forces and can thus effectively describe biaxial bending in RC columns. Shear effects are also easily modeled in two-node force-based elements. The formulation of a force-based Timoshenko beam is very similar to that of an EulerBernoulli beam. At the section level, different approaches can be used. The most general one is to couple axial, bending and shear deformations at the section level, but this requires sophisticated cyclic material laws. Petrangeli et al. [19] have proposed a fiber element with shear deformations for the uniaxial bending case. An alternate formulation decouples the section shear force from the bending and axial forces in the section constitutive law (that is an independent, uncoupled shear V-γ law is assigned to the section), and equilibrium at the section level couples bending and shear responses. In other words, when the shear strength is reached, bending cannot increase to equilibrium between shear forces and bending moments [14]. It can however be stated that a section model that accounts for full nonlinear coupling between axial, biaxial bending, shear and torque is not yet available for beam elements. Different uniaxial constitutive laws are available for concrete and steel. For concrete, different properties can be assigned for confined and unconfined concrete. Modifications of the steel bar laws can also account for buckling and for slip, but calibration is a key issue in this case. At present, no readily available models are yet available for the beam-column joints, which are commonly neglected or assumed rigid in current practice. Another important issue is modeling the effects of the frame infills. It is well known that masonry infills influence the behavior of RC frames, but modeling the interaction is not easy. Strut elements have been proposed, but the selection of the material properties is not straightforward. As for nonlinear geometric models, there is little experience on their use in nonlinear frame analysis, even though many think it is important as the damage induced by the ground motions leads to larger lateral drifts. For displacement-based elements, treatment of the P-∆ effects or even of the large-displacements and moderate rotations is a well established field. For 8


force-based elements, the problem has been recently solved ([18][23]) within the general corotational framework. Spread plasticity models are no doubt more precise and accurate than lumped-plasticity models, However, they are somehow more delicate to use. For section behaviors that exhibit softening (reinforced concrete columns, local buckling etc.) spread plasticity elements localize the deformations in the extreme sections and the solution loses objectivity ([7]). Several solutions have been proposed, all based on the knowledge of the actual plastic hinge length Lpl. This is however a central point to the entire nonlinear procedure. The analyses are based (for both lumped and spread plasticity models) on the notion of a plastic hinge length. Yet, there does not yet exist a unified definition of this characteristic length, because the problem is not totally understood or described from a mechanical viewpoint. In other words, nonlinear analyses describe a process, the formation and evolution of the plastic hinge, that is not fully understood. It is therefore difficult to give reliable formulas for Lpl. Though EC8 provides some indications on how to compute the plastic hinge lengths, these definitions seem to apply for new buildings where the rebar splicing is placed away from the member ends and the end regions are well confined by the transverse reinforcement. For existing buildings, the plastic hinge length may be lower. The application of more advanced methods of analysis such as the fiber elements is sometimes incompatible with the nonlinear procedures specified by the design codes. This is precisely the case of the N2 method in EC8. In this case, for plan-irregular buildings, one has to perform two different pushover curves in the principal x and y directions of the building. The results are then combined using the combination rules of Figure 3. When using the fiber element, though, a problem arises. Both pushover analyses are performed after application of the gravity loads. But the gravity loads cannot be accounted for twice, thus the procedure of Figure 3 can only be rationally applied with section laws that consider axial and bending forces uncoupled. The problem is more general. The combination rules developed for linear methods do not readily extend to nonlinear methods, because the principle of superposition no longer applies. The main problem here is that the N2 method was developed for 2d models, and its extension to 3D models is quite difficult and is becoming unrealistic. Another problem that arises when using fiber element has to do with the selection of the control point. The original N2 method was developed for shear frames and can be easily applied to frames with floor diaphragms which are rigid in their plane. When the beams are modeled with fiber elements, floor diaphragms cannot be used, because they introduce artificial compression due to the fact that the beams cannot freely extend. Because the beams are normally axially stiffer, almost any point can be selected. However, in older buildings with few beams and thin slabs the floor can become quite flexible in its plane and thus some differences can be expected in the horizontal displacements over a floor. Also, slabs have to be modeled with either shell elements or with equivalent bar elements (which ignore the flexural stiffness of the slabs). NTH analyses appear to be more reliable than NPO analyses, but some problems exist in their case too. Slowness of the analysis was already mentioned. Another issue that is sometimes neglected is the direction along which the ground motion is applied. Design codes typically apply the ground motions (either in the form of design spectra or of input accelerograms) along the structure principal directions. Such directions may be difficult to identify for older, plan irregular buildings or for skewed bridges. The problem does not exist for linear analysis, where the directional combination rules (CQC3, SRSS, 100/30 etc.) cover the possibility of different ground motion direction. No thorough studies have been completed for nonlinear analyses of plan-irregular structures.

9


4

EXAMPLES

Some of the previously highlighted issues are now discussed through the nonlinear analyses of a RC building and a RC bridge. The RC building is a 3-story structure that was damaged during the 2002 Molise earthquake in Italy. It is a typical residential building with a semi-open ground level. The scope of the study, conducted jointly between the University of Chieti-Pescara, Italy and the University of California, San Diego, USA, is to assess the effects of different modeling details on the structural response of the building model.

z y

x

Figure 4 Bonefro building

The building is analyzed using OpenSEES ([15][16]). The basic frame is modeled with force-based, fiber-section distributed plasticity beam elements (schematically shown in Figure 5. The general section model uses the section aggregator in OpenSEES, where shear is coupled with bending only at the element level. In the applications described in this paper, only linear shear and linear torque were used. One element per structural element is used, unless there is a change in section reinforcement. The bare frame response is shown in Figure 7. The bare frame is in this case loaded in the global Y direction only. The structure is particularly flexible in the Y direction because the central frames have no beams. A 1.18sec fundamental period of the frame structure in the Y direction is found from a modal analysis after the application of the gravity loads. σ

Nonlinear Force-Based element (distributed plasticity)

ε

σ ε Figure 5 Frame element, fiber section and fiber steel and concrete laws used NL fiber sctions

As previously noted, it is not clear how the N2 Pushover procedure can be applied in this case where a fiber section model is used. The application of the procedure of Figure 3 implies two pushovers in the X and Y plan principal directions. Does that imply that the gravity loads are applied twice? If they are applied twice, then the resulting axial loads will be overestimated, if the gravity loads are applied only once, before say the pushover in the X direction, then the N-M interaction in the pushover in the Y direction is not very meaningful. It appears

10


that in the current format the N2 procedure cannot be applied to irregular buildings where the structure has to be loaded in two orthogonal directions. fiber NL fibre

b

b

+

h

σ

torque L or NL T- θ

shear L or NL V- γ

σ-εε

+

h

σ

V

ε

b

T

GAs

ε

Kt θ

γ KT = V

GJ Gc2 hb = L L

3

(h > b)

P, M, V, T

Mt

γ

P, M

θ

h

θ

V

Mt

Figure 6 Extended section model (section aggregator in OpenSEES [15])

The bare frame of Figure 7 was studies using both modal and uniform pushover according to EC8, modal pushover and nonlinear time history analysis using seven generated accelerograms. The green curves in Figure 7 indicate the responses to the seven accelerograms, while TH indicates the average value of the maximum response quantity, as prescribed by EC8. The results of the pushover analyses are in good agreement with the average time history displacement response, while some inaccuracy appears in the upper floors drift response. The MPA analysis results show that the higher modes have some participation, but their contributions remain essentially elastic. The SRSS combination rule used in this case leads to a total MPA response slightly greater than the mode 1 only PO response. It is noteworthy that all three methods agree in finding a floor mechanism at the second story. Due to the long fundamental period of the structure, the total response has a large elastic component. The plastic hinge inelastic demand varies from column to column depending on the presence of adjacent beams and on the height of the beams. The floor is typically modeled as a diaphragms rigid in its plane. This may not be a good choice in the present example because: a) the absence of beams in the central frames makes the floor flexible; b) the rigid diaphragms impose artificial prestressing in the fiber-section beams by blocking their extension. A 2x2 mesh of 40mm-thick elastic 4-node concrete shell elements is used for each slab field (Figure 11). Similarly to the previous case, the average TH response and the first mode PO analysis match quite well in terms of absolute displacements, while the PO drift demand is still lower. There is also some increased scatter among drift responses obtained with different ground motions at the floor where the maximum drift occurs. As far as the structural behavior is concerned, the presence of the floor slabs unloads the beams with respect to the previous case. The additional floor bending stiffness produces lower elastic deformations in the columns. The elastic configuration of the floor slab somehow bounds the slab behavior since slabs are expected to crack. The next element to be added are the infills. The infill panels are modeled as double inelastic no-tension struts, with thickness tw=200mm, equivalent width dw=d/10, using a uniaxial 11


elastic, softening law similar to that of concrete. Two cases are considered. In the first, the struts connect the structural joints, in the second, at the ground level the struts are connected to the columns at a distance of approximately 500 mm from the beam-column joint (Figure 9). 4 2 0.2

5 0.2

TC

0. 4

0.1

5

1 0.9

Modal Pushover (MPA) Capacity-demand spectra

3

0.7 0.6

Floor

*

Se/g , F /gm

*

0.8

0.5

TH POunif Mode1 Mode2 Mode3 MPA

2

0.4 4 1.3

0.3

1

0.2

2

0.1

0

0 0

0.05

0.1

0.15

0.2

0

50

2000

150

200

4

Pushover curves 1500 Mode2

POunif

3

1000 Mode3

Floor

Base Shear (KN)

100

Displacements (mm)

Sde (m)

Mode1

500 0

2

1

0

50

100

150

200

Top floor displacement (mm) 0

0

0.5

1

1.5

2

Drift %

Figure 7 Results of NPO on Bonefro building (bare frame) 4

Floor

3

2

1

0

TH POunif Mode1 0

50

100

150

200

Displacements (mm)

3

2000 Pushover curves 1500

Floor

Base Shear (KN)

4

POunif

1000

Mode1

500

2

1

0 0

50 100 150 Top floor displacement (mm)

200

0

0

0.5

1

1.5

2

Drift Ratio %

Figure 8 Results of NPO on Bonefro building (bare frame plus flexible floors)

The first effect is to increase the stiffness of the building and the base shear in the building, but to decrease the total displacement demand. The pushover response is in good agreement with the time history average results in terms of maximum top displacements. Some differences are found in the maximum drift response at the upper floors. For the case of the eccentric struts, the base column shear behavior was modeled with a nonlinear shear law. The overall response does not change significantly, but the central column of the lateral frames fail 12


2000

Base Shear (KN)

Base Shear (KN)

in shear. The shear failure is considered using the model of Figure 6 with a nonlinear shear behavior. The central base column response is shown in Figure 10. When the shear strength is reached (upper right plot), by equilibrium the bending moment (still mostly elastic, lower right plot) cannot increase.

POunif

1500

Mode1

Pushover curves

1000 500 0 50 100 150 Top floor displacement (mm)

Mode1

1000 500 0 0

200

4

4

3

3

2 TH POunif Mode1

1

0

0

50

100

150

50 100 150 Top floor displacement (mm)

2

1

0

200

0

50

100

150

200

Displacements (mm) 4

4

3

Floor

3

Floor

200

TH POunif Mode1

Displacements (mm)

2

2

1

1

0

Pushover curves

POunif

1500

Floor

Floor

0

2000

0

0.5

1

1.5

0

2

0

0.5

Drift Ratio %

1

1.5

2

Drift Ratio %

Figure 9 Results of NPO on Bonefro building (bare frame plus infills)

2°level

Infill strut

Shear Z (KN)

200 100 0 -100 -200 -0.02

-0.01

0

0.01

0.02

0.1

0.2

Moment Y (KNm)

Gamma Z

Short column 1°level

200 100 0 -100 -200 -0.2

-0.1

0

Curvature Y (1/m)

Eccentric Infill strut

Figure 10 Response of the central base columns with eccentric infills

Finally, the bare frame was studied using two accelerograms applied in the X and Y directions. Figure 11 shows the bending moment response at the base section of the highlighted columns for one accelerogram applied in the Y direction only, while Figure 12 shows the same response due to simultaneous accelerograms in the X and Y directions. The section re-

13


sponses change, mainly due to the change in axial load due to the application of the accelerogram in the orthogonal direction. In general, the change in axial load significantly affects the bending response. The outer column is more loaded because there are no beams in the Y direction.

Moment (KNm)

200

0

-200

-0.1

-0.05

0

0.05

0.1

0.15 Moment (KNm)

200

0

-200

-0.1

-0.05

0

0.05

0.1

0.15 Moment (KNm)

200

0

-200

-0.1

-0.05

0

0.05

0.1

0.15

200

Moment (KNm)

Moment (KNm)

Moment (KNm)

Moment (KNm)

Moment (KNm)

Mx, ay

0

-200

-0.1

-0.05 0 0.05 Curvature (1/m)

0.1

0.15

200

0

Z

-200

-0.1

-0.05

0

0.05

0.1

0.15

200

0

-200

-0.1

-0.05

0

0.05

0.1

0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.1

-0.05 0 0.05 Curvature (1/m)

0.1

0.15

200

ay

Y X Columns 1008 1011 j

0

-200 200

0

-200

i

Figure 11 Results of NTH on Bonefro building (bending moment Mx due to accelerogram ay only)

Moment (KNm)

200

0

-200

-0.1

-0.05

0

0.05

0.1

0.15 Moment (KNm)

200

0

-200

-0.1

-0.05

0

0.05

0.1

0.15 Moment (KNm)

200

0

-200

-0.1

-0.05

0

0.05

0.1

0.15

200

Moment (KNm)

Moment (KNm)

Moment (KNm)

Moment (KNm)

Moment (KNm)

Mx, ay+ax

0

-200

-0.1

-0.05 0 0.05 Curvature (1/m)

0.1

0.15

200

0

-200

Z -0.1

-0.05

0

0.05

0.1

0.15

200

0

-200

-0.1

-0.05

0

0.05

0.1

0.15

-0.1

-0.05

0

0.05

0.1

0.15

-0.1

-0.05 0 0.05 Curvature (1/m)

0.1

0.15

200

Y X Columns 1008 1011 j

ax+ay

0

-200 200

0

-200

i

Figure 12 Results of NPO on Bonefro building bending moment Mx due to accelerograms ax and ay)

The second example is a RC bridge located in Vasto Marina (south of Pescara, Abruzzo, Italy) along the major north-south Adriatic Highway 16. It is a two-lane bridge built in the

14


1960s. The bridge, shown in Figure 13, has a sharp bent about 1/3 from the south end. The supporting piers are multi-column bents. The bridge is located near the Adriatic seacoast and some of the steel reinforcement is badly rusted. The bridge is irregular in plan and height.

Front view

Bari

P esca

ra

Plan

1,0 4,50 2,0

10,20

17,45

Longitudinal section

Transversal section

Figure 13 Vasto Marina, Italy Viaduct Roller (Equal constrain)

Support (Equal constrain)

Constrain yz

z

y x

Constrain xz

Figure 14 Bridge details and analytical model

The superstructure is partly continuous, with Gerber beams connecting the continuous sections of the bridge. The bridge has an irregular geometry. Its complex dynamic response can15


not be captured by analyzing separate subsystem. Furthermore, the Gerber supports are rollers in the longitudinal direction only, while they block the transversal displacements. For these reasons, a global model of the entire bridge was created placing particular care in the definition of joints and the nodes. The pier bents are modeled with beam elements, whereas the deck is modeled with shell elements as shown in Figure 14. The Gerber support joints are modeled with two overlapping nodes connected with equal displacement constrains in the x, y and z direction (equal constraints). The roller joints are modeled in a similar manner but the x direction is not constrained. To simulate correctly the flexural behavior of the piers in the longitudinal and transversal direction, constrains are inserted in the nodes as shown in Figure 14 The bridge is studied with linear and nonlinear dynamic analyses, using the Italian draft guidelines for the seismic vulnerability assessment of existing bridges, prepared in the framework of the Reluis project, funded by the Italian Civil Protection Department in support of the new seismic design guidelines [19]. The linear analysis is the classical response spectrum analysis (RSA) according to EC8. The nonlinear static analysis is performed with lumped plastic hinges and the response time histories (RTH) are performed with fiber force based elements. The results of the RTH performed with the program Midas Civil [16] are reported here. To perform the analysis, 2 accelerograms are selected. The accelerograms are part of a suite of ground motions that comply with the requirements of the new Italian seismic codes, OPCM 3431 [19], They can be found at the website: http://reluis.rdmweb.com/pagine/Accelerogrammi_europei.htm. The structure sits on type C soil (OPCM 3431 [19]). The material properties used in the analysis are the following: concrete cylinder strength, fc=20 MPa, steel yielding strength fy=315 MPa. The concrete is modeled using the Kent and Park model, whereas the steel is modeled using the Menegotto-Pinto model. The sections are subdivided into 10x10 fibers. Because program Midas does not included tapered elements, the tapered piers are idealized with two elements and every element is integrated with 3 Gauss-Lobatto points. In the longitudinal direction, the pier height varies between 3 (Bent 1) and 10 (Bent 14) meters, therefore the length of the first Gauss-Lobatto point varies between 250 mm and 837 mm. The base section dimensions of bent 1 are h=700 mm and b=350mm, whereas the base section dimensions of bent 14 are h=880 mm and b=350 mm. The design base acceleration for the site is ag = 0.15g. Five incremental dynamic analysis are performed with ag = 0.15g, 0.12g, 0.09g, 0.06g and 0.03g. The shear strength is checked using the Eurocode 2 (2005), which uses the modified field theory. The results are represented in Figure 15. The bent are numbered from the left to the right from 1 to 14.

Vy Vy

Vxy Vy Vy

Vxy Vy Vy

Vy

ag = 0.06g

Vy

ag = 0.09g

V V V Vy Vxy Vxy Vy Vy Vy Vxy Vy y y y

Vy Vy Vy Vy

Vy

ag = 0.15g

ag = 0.12g Figure 15 Shear failures for different PGAs

In Figure 15 Vx indicates shear failure in the longitudinal direction and Vy shear failure in the transversal direction. For ag = 0.03g the piers do not collapse. The first column to collapse 16


in the longitudinal direction is in bent 1 for an acceleration ag = 0.06g. The base shear of the first column that collapses is depicted in Figure 16 for different levels of the PGA.

Base acceleration

0.15 0.12 0.09 Vy Vx

0.06 0.03 0 0

100

200

300

400

Base Shear [kN]

Figure 16 Pier 1 base shear force

For an acceleration of 0.15 g nearly all the elements fail in shear. The members fail in flexure when the ultimate concrete strain is reached. Figure 17 shows the deformed shape when the first column collapse in flexure.

Acc. [m/s2]

1.5

Failure 0

-1.5 0

4 Time [s]

F

Y

8

Y

12

Y

Y

YYYY F

YYYYY

Figure 17 Deformed shape at flexural failure (7.78 seconds)

In the Figure 17 F stands for flexural failure and Y indicates yielding of the rebars. The picture clearly indicates that the high torsion of the first bent causes failure of the outmost external column. The base section at failure is shown in Figure 18. There is a high biaxial bending which causes the neutral axis to rotate. The moment-curvature behavior in the two orthogonal directions is shown in Figure 19. It is worth noting that in several codes the ultimate capacity is estimated on the basis of the ultimate chord rotation. For example Eurocode 8 uses the following formula:

 L  θ p ,u = ( Φu − Φ y ) Lpl  1 − pl   2L 

(3)

where Φu is the ultimate curvature, Φy is the yielding curvature, Lpl is the plastic hinge length and L is the distance from the end section of the plastic hinge to the point of zero moment in the pier (shear span length). The plastic hinge length Lp can be found as:

17


Lpl = 0,10 LV + 0,015 f yk d bL

(4)

The formula was calibrated on the basis of simple bending tests. Is it reliable for biaxial bending? Moreover, how can LV be defined in the case of biaxial bending? It appears that the literature published and the guidelines available at present do not investigate in depth this problem. Also Eq. (4) is applicable to well confined sections only, thus it may not be readily extended to unconfined sections, as is the case of the given example. -4.00

εconc > 0.002

-2.00

εs > εy

0.00 0

2.00

4.00

n.a

Stress [MPa]

-5 -10 -15

.

-20 Strain x 1000

εconc = εcu = 0.0035

Figure 18 Base section at failure

500 N=1098 kN

M [kNm]

250 Mz-Rz

-0.02

-0.01

0 0.00

0.01

0.02

-250 N=1098 kN

My-Ry -500 R [1/m]

Figure 19 Moment-Curvature at the bottom point

5

CONCLUSIONS

It is recognized that nonlinear methods of analyses are the rational procedures for predicting the response of structures under moderate and strong earthquakes. Much work and progress have been achieved in recent years, in terms of both model and nonlinear procedure developments, but nonlinear methods of analyses, as proposed by EC8, are not yet routinely used in either research or practice. The following is a partial list of issues that still need clarification: • The nonlinear pushover procedure is not readily applicable to plan irregular buildings or structures. This simplified and quick method of analyses was originally developed for plane frame and its extension to space frame is not straightforward and may not be achievable; • The selection of the ground motions for nonlinear dynamic analyses is key to these

18


analyses but no clear procedure is yet in place. Another issue that needs further investigation is the amount of viscous damping that can be assigned. Finally, the directions along which the ground motions are applied to plan-irregular buildings is easily identified; • The differences found with different modeling assumptions need to be fully understood by the analysts. Floor flexibility, infill effects, shear response are some of the effects that the analyst may want to consider; • Not all nonlinear phenomena (such as joint failures, frame-infill interaction and shearbending interaction) are modeled correctly by present nonlinear computer codes, either research or commercial; • The amount of output produced by nonlinear computer codes is very large. Though graphic post-processing tools are available (mainly in commercial codes) the design checks required by EC8 need further testing before they are fully applicable. Once clear procedures are identified, then post-processing of the results will be easily implemented in the nonlinear codes.

As more and more structures are analyzed with nonlinear procedures, the above issues will become clearer and the problems will be solved, especially through a strong interaction between practicing engineers and researchers.

REFERENCES

[1] ATC 40, Seismic Evaluation and Retrofit of Concrete Buildings, Applied Technology Council, 1996 [2] A.S. Elnashai, Do we really need Inelastic Dynamic Analysis?, Journal of Earthquake Engineering, 6, Special Issue 1, 123-130, 2002. [3] Eurocode 8: Design of Structures for Earthquake Resistance, European Committee for Standardization, 2003. [4] S. Antoniou, R. Pinho, Advantages and Limitations of Adaptive and Non-adaptive Force-Based Pushover Procedures, Journal of Earthquake Engineering, 8(4), 497-522, 2004. [5] M.N. Aydinoglu, An Incremental Response Spectrum Analysis Procedure Based on Inelastic Spectral Deformation for Multi-Mode Seismic Performance Evaluation, Bulletin of Earthquake Engineering, 1, 3-36, 2003. [6] A.K. Chopra, R.K. Goel, A modal pushover analysis procedure for estimating seismic demands for buildings, Earthquake Engineering and Structural Dynamics, 31, 561-582. 2002. [7] J. Coleman, E. Spacone, Localization Issues in Nonlinear Force-Based Frame Elements, ASCE Journal of Structural Engineering, 127(11), pp. 1257-1265, 2001. [8] P. Fajfar, Capacity Spectrum Method Based on Inelastic Demand Spectra, Earthquake Engineering and Structural Dynamics, 28, 979-993, 1999. [9] P. Fajfar, Structural Analysis in Earthquake Engineering – A Breakthrough of Simplified Non-Linear Methods, 12th European Conference on Earthquake Engineering, London, United Kingdom, Paper 843, 2002.

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[10] P. Fajfar, V. Kilar, D. Marusic, I. Perus, The extension of the N2 method to asymmetric buildings, 4th European Workshop on the Seismic Behaviour of Irregular and Complex Structures, Paper No. 41, Thessaloniki, Greece, 2005. [11] FEMA 356: Prestandard and Commentary for the Seismic Rehabilitation of Buildings, Prepared by American Society of Civil Engineers, Washington, D.C. (U.S.A.), November 2000 [12] B. Gupta, S.K. Kunnath, Adaptive Spectra-Based Pushover Procedure for Seismic Evaluation of Structures, Earthquake Spectra, 16(2), 367-391, 2000. [13] I. Iervolino, G. Maddaloni, E. Cosenza, Unscaled real record sets compliant with Eurocode 8. First European Conference on Earthquake Engineering and Seismology, Geneva, Switzerland:2006 [14] A. Marini, E. Spacone, E. “Analysis of R/C Elements Including Shear Effects.” ACI Structural Journal, 103(5), 645-655, 2006. [15] McKenna F. (1997), Object-oriented finite element analysis: frameworks for analysis, algorithms and parallel computing. Ph.D. dissertation, University of California, Berkeley. [16] Fenves G., McKenna F., Scott M.H., Takahashi Y.: An Object-Oriented Software Environment for Collaborative Network Simulation, 13th World Conference on Earthquake Engineering, Paper 1492, Vancouver, B.C. (Canada), 1-6 August 2004. [17] MIDAS, Civil Analysis Reference Manual, Civil 2006 (v7.12 Release No.1) Midas Information Technology Co. Ltd., 2006 [18] A. Neuenhofer, F.C. Filippou, Geometrically Nonlinear Flexibility-Based Frame Finite Elements, ASCE Journal of Structural Engineering, 124(6), 704-711, 1998. [19] OPCM 3431 – Ulteriori modifiche ed integrazioni all’OPCM3274, 2005 (in Italian, Technical Norms for design, evaluation and seismic rehabilitation of buildings). [20] M. Panagiotou, J. Restrepo, Computational model for the UCSD 7-story structural wall building slice, SSRP 07-09 Report, Department of Structural Engineering, University of California San Diego, 2007 (in press). [21] M. Petrangeli, P.E. Pinto, V. Ciampi, Fiber element for cyclic bending and shear of R/C structures. I: theory, ASCE Journal of Engineering Mechanics, 125(9), 994-1001, 1999. [22] M.H. Scott, G.L. Fenves, Plastic Hinge Integration Methods for Force-Based BeamColumn Elements, ASCE Journal of Structural Engineering, 133(2), 244-252, 2006 [23] M.H. Scott, F.C. Filippou, Response Gradients for Nonlinear Beam-Column Elements under Large Displacements, ASCE Journal of Structural Engineering, 133(2), 155-165, 2007. [24] E. Spacone, F.C. Filippou, F.F. Taucer, Fiber Beam-Column Model for Nonlinear Analysis of R/C Frames. I: Formulation, II: Applications. Earthquake Engineering and Structural Dynamics, 25(7), 711-742, 1996. [25] Th.Zimmermann, A.Truty, A.Urbanski, K.Podles. Z-Soil user manual, Zace Services Ltd, 19852007

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