# Derivation of fragility curves for RC frames retrofitted with RC infill walls based on full-scale pseudodynamic testing results

Christis Z Chrysostomou, Nicholas Kyriakides, Panagiotis Kotronis, E Georgiou

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Christis Z Chrysostomou, Nicholas Kyriakides, Panagiotis Kotronis, E Georgiou. Derivation of fragility curves for RC frames retrofitted with RC infill walls based on full-scale pseudodynamic testing results. VII European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS Congress 2016 ), Jun 2016, Limenas Chersonisou (Crete), Greece. hal-05480652

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
DERIVATION OF FRAGILITY CURVES FOR RC FRAMES
RETROFITTED WITH RC INFILL WALLS BASED ON FULL-SCALE
PSEUDODYNAMIC TESTING RESULTS

C.Z. Chrysostomou<fnref n="1" />, N.C. Kyriakides<fnref n="1" />, P. Kotronis<fnref n="2" /> and E. Georgiou<fnref n="1" />

e-mail: c.chrysostomou@cut.ac.cy

e-mail: nicholas.kyriakides@cut.ac.cy

e-mail: e-mail: es.georgiou@edu.cut.ac.cy

e-mail : Panagiotis.Kotronis@ec-nantes.fr

**Keywords:** Retrofitting of RC buildings; RC Infill Walls; Connection details; finite element analysis; fibre model, pseudo-dynamic experimental results

**Abstract.**

The effectiveness of seismic retrofitting of RC-frame buildings by converting selected bays into new walls through infilling with RC walls was studied experimentally using a full-scale four-storey model tested with the pseudo-dynamic (PsD) method. The frames were designed and detailed for gravity loads only using different connection details between the walls and the bounding frame. In order to simulate the experimental response, two numerical models were formulated differing at the level of modelling. The initial purpose of this paper is to illustrate the capabilities of these models to simulate the experimental nonlinear behaviour of the tested RC building strengthened with RC infill walls and comment on their effectiveness. In the second part of the paper vulnerability analysis of the strengthened frame was undertaken in order to derive fragility curves for buildings retrofitted with the specific retrofit method. The experimental results of the floor displacements are correlated with the propagation of damage observed during the experiment in order to derive a framework for hybrid fragility curves. Since the simple analytical model was found to simulate closely the response of the retrofitted frame, it was used for the derivation of the fragility curves. The comparison of the fragility curves of the pre and post retrofit building provides important quantitative information regarding the decrease in probability of severe damage and collapse. The improvement in the dynamic properties of the building after the retrofit are also incorporated in the framework and a discussion is undertaken regarding the effectiveness of retrofitting of RC buildings through the insertion of RC walls.

<footnote n="1">¹ Cyprus University of Technology, Department of Civil Engineering and Geomatics, P.O.Box 50329, 3603 Lemesos,
Cyprus</footnote>

<footnote n="2">² LUNAM Université, Ecole Centrale de Nantes, Université de Nantes, CNRS UMR 6183 GeM (Institut de Recherche en Génie Civil et Mécanique), 1 rue de la Noé, BP 92101, 44321 Nantes, France</footnote>
# 1 INTRODUCTION

The most effective and economic method for retrofitting multi-storey RC buildings is the construction of new walls, especially those with pilotis (soft-storey). Their structural and economic effectiveness increases when selected bays of an existing RC frame are fully infilled. Most of the experimental research work performed in the last decades covers sufficiently the other frequently used types of retrofitting – in particular the use of fibre-reinforced polymers (FRP) and the concrete jackets. Some of these types of retrofitting for RC buildings with openings are discussed in Hemant (2009). Some more recent work (Antonopoulos T.A and Anagnostopoulos S.A., 2012) examines numerically the effectiveness of the use of diagonal steel braces to provide stiffness and ductility in cases of open ground stories. It is shown that the application of these braces improves the demand to capacity ratios for shear and bending in all floors of the building.

Alternatively, Misir (2014) proposes the use of locked brick infill walls, which have the potential to decrease the soft-story/weak-story formation in comparison to standard bricks due to their shear sliding mechanism and decreased upper-story/first-story stiffness, even in buildings that have non-infilled first stories. The use of these sliding brick walls increases the capacity of the building since it removes weaknesses due to openings but cannot provide significant stiffness and strength to the building to withstand severe earthquakes.

The use of RC Infill walls for retrofitting RC buildings with openings is a relatively new retrofit method, which can be used to increase the strength, stiffness and ductility of the building. There is no adequate experimental research work on the use of RC infill walls and most research has mainly targeted on what is feasible: testing of one- to two-storey specimens due to the practical difficulties of testing large specimens with high resistance. So, data is lacking for taller full-scale specimens that reflect real applications. Regarding to code provisions, Eurocode 8 – Part 3 fully covers retrofitting with FRP or concrete jackets, while it does not refer at all to new walls created by infilling frame bays. Other guidelines, like KANEPE (2012) in Greece, refer to the design of such walls but only in terms of forces, without providing tools for calculation of their characteristic deformations (at yield and failure) and stiffness, unless the infill wall can be considered integral with the bounding frame. The inadequacy of design codes in this respect is due to our poor knowledge of the behaviour of walls created by infilling with RC a bay of an existing frame.

In order to start filling the gap of knowledge regarding infilling of existing RC frames with RC walls, the effectiveness of seismic retrofitting of multi-storey multi-bay RC-frame buildings by converting selected bays into new walls through infilling with RC was studied experimentally at the Elsa Laboratory of Structural Assessment (ELSA) facility at JRC, in Ispra. The research was under the project "Seismic Engineering Research Infrastructures for European Synergies" (SERIES), financed by the Seventh Framework Programme of the European Commission.

The scope of this paper is the numerical simulation of the tested frame and the comparison of the results with the experimental ones, using numerical models of different modelling sophistication. In the first part, a brief description of the design of the bare-frame specimen and the RC Infill walls is presented along with details of the testing campaign and some of the main results obtained from the experiment. Details can be found in Chrysostomou et al. (2013 and 2014) and Poljansek et al. (2014). In the second part, which is the main focus of the paper, the modelling and the results a 2D finite element micro-model, simulating the behaviour of one of the two planar frames tested, and a macro-model using fibre elements for the frame members and shell elements for the walls, simulating both planar frames and the connecting beams and slab, are presented. Finally, the results of the numerical simulation of the bare
frame are also provided in order to emphasise the effectiveness of this retrofitting method and conclusions are drawn.

## 2 DESCRIPTION OF TEST

The specimen was designed based on a four-storey prototype building structure consisting of four three-bay frames spaced at 6 meters, with RC infilling of the exterior frames only. The specimen was designed at full-scale to represent the two exterior frames of the prototype structure, spaced at 6 m and linked by 0.15 m thick RC slab and four 0.25 m by 0.50 m beams perpendicular to the plane of the two three-bay frames. The purpose of the experiment was to perform uni-directional pseudo-dynamic testing on these two parallel planar frames, which were infilled with RC infills reinforced with different amounts of reinforcement and connected to the frames with different connection details, as explained in the following paragraphs. These frames were named North and South as it is defined in Fig. 3. Hence, the direction towards the reaction wall (backside of Fig. 3) is East and the one in the opposite direction (front-side of Fig. 3) is West. The centre-line length dimension of the specimen was 8.5 m (central bay 2.5 m and the two exterior bays 3.0 m), the storey height 3.0 m, and the total height of the specimen (excluding the foundation) 12.0 m (Fig. 1).

The dimensions of the columns were 0.25 m by 0.40 m with the long dimension along the plane of the three-bay frames (that was also the direction of uni-directional loading), while those of the beams 0.25 m by 0.50 m.

<figure><img src="image_2.png" /><figcaption>Figure 1. Dimensions of the full-scale specimen without the RC infill</figcaption></figure>

The proposed structure represents typical construction of the late 70's and beginning of the 80's in Cyprus. At that time, there were no provisions for earthquake loading, so the structures were designed for gravity loads only. There was no specific design standard and the authorities were accepting any standard that was acceptable to other countries such as CP110 (BSI, 1972) and BS8110 (1983), DIN, Greek Code, US code etc. For the mock-up design, it was decided to use the provisions of BS8110, which were very close to those of CP110 with minor differences.

The material properties used in the mock-up were constrained by the availability of materials in the Italian and European market. It was finally decided to use concrete C20/25 for both the frame and the walls, of unit weight 25 kN/m³ and modulus of elasticity, E=30,000 MPa. The yield strength of the ribbed bar reinforcing steel was fyk = 400 MPa for both bending and shear reinforcement of the frame members and the slab, while for the RC infill and the dowels to be used for connecting the wall to the bounding frame members the yield strength was specified to fyk = 450 MPa. The 400 MPa characteristic yield strength steel represents the one used in Cyprus construction practice in the 1970's and 80's, while the 450 MPa was the clos-
est available in the Italian market to substitute for the 500 MPa steel that would have been
used in the walls for retrofitting such a structure.

The self-weight was calculated using the unit weight of concrete specified above. The im-
posed dead load including the load of masonry infill walls was 3kN/m2, and the live load was
1.5kN/m². These loads were combined using partial factors of safety of 1.4 for self-weight
and imposed dead-load, and 1.6 for the live load. The material partial factors used were 1.5
for concrete and 1.15 for steel.

The reinforcement details of the beams, the columns and the slabs are given in Chrysosto-
mou et al (2013 and 2014) and Poljansek et al. (2014). Herein, it is important to emphasise the
fact that a compression lap was used for columns, in line with a design for gravity loads only,
and was expected to fail when subjected to tension. Connection details at the joints, laps and
between the infill-walls and the bounding frame are given in Fig. 2.

<figure><img src="image_3.png" /><figcaption>Figure 2. (a) Reinforcement details for beams, (b) Reinforcement details for columns, (c) Dowels and starter bars, (d) Dowels, starter bars and web reinforcement</figcaption></figure>

The walls in the two frames, which had a thickness of 0.25 m equal to the width of the
beams and columns of the bounding frame, were reinforced with different amounts of rein-
forcement, with the north one being the stronger of the two, in order to facilitate the study of
the effect of as many parameters as possible.

Two parameters were examined: a) the amount of web reinforcement in the walls and b) the connection detail between the wall and the bounding frame. Regarding the connection detail, two distinct connection details were used. In the first detail, the web bars were connected to the surrounding frame through lap splicing with the same diameter starter bars epoxy grouted into the frame members. Short dowels were then used in order to transfer the shear at the interface between the wall and the frame member. This detail was used to connect the wall at the bottom beam and right column at the 1st and 2nd floors of the North frame (Fig. 2c, starter bars are shown here only for the bottom beam), while for the South frame it was used to connect the wall at the bottom beam of the 1st and 2nd floors, and the west and east columns of the 1st and 2nd floors, respectively.

In the second detail, longer dowels were used to act both as dowels and as starter bars of
the web reinforcement, which is placed at the two faces of the wall. Thus, these dowels are
considered as lap-spliced with the nearest – smaller diameter – web bars. However, in this
case, the clear distance between the dowel and the nearest web bar, violates the maximum
clear distance of 50 mm or 4Φ between lapped bars, specified in Eurocode 2 (CEN, 2004).
This detail was used to connect the wall at the top beam and west column at the 1st and 2nd
floors of the North frame (Fig. 2c), while for the South frame it was used to connect the wall
at the top beam of the 1st and 2nd floors, and the east and west columns of the 1st and 2nd
floors, respectively. In the 3rd floor of both the North and South frames, only the second detail
was used, while for the 4th floor only two dowels per wall interface were used to provide
safety against falling of the wall out of plane. The completed wall reinforcement (including
web, starter bars and dowels) for the 1st floor of the North wall is shown in Fig. 2d. In all
cases, the dowels were positioned along the centreline of the elements (i.e. at 0.125 m from
the face of the wall). Detailed guidelines for the design of such walls can be found in Fardis (2013).

Since the lapping of the column reinforcement could take only compression, then it was obvious that there would be lap splice failure at the ground floor columns next to the RC Infill wall, which could be detrimental to the whole experiment. Therefore, in order to safeguard against this type of failure and allow the experiment to be performed without any premature failure, it was decided to reinforce the edges of the wall at the 1st floor with three-sided CFRP for the height of 0.60 m (lapping stopped at 0.55 m) from the base of the column. Based on the performance of the ground floor RC Infill wall during the experiments, this additional retrofit of the lap length region is crucial and constitutes part of the proposed retrofit strategy.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Elevation of the specimen in the lab. The wall shown on the right is the South wall (frame) and the one on the left is the North wall (frame)</figcaption></figure>

Within the testing campaign two PsD tests and one cyclic test were run. The accelerogram was scaled to a maximum acceleration of 0.10g and 0.25g for the first and second PsD tests, respectively. For the final cyclic test, a history of displacements was imposed at the fourth floor, while maintaining a triangular distribution of loads along the height of the North frame and zero rotation at each of the four floors (unidirectional test). The aim of this final test was to explore the final capacity of the specimen up to a 20 % drop of peak strength of the structure.

In a PsD test, it is assumed that the response of a structure can be determined by a discrete model with a limited number of degrees of freedom (DoF). In this test campaign, four DoFs were selected: the horizontal displacements of each storey with the assumption that all the mass is concentrated at the selected DoFs (i.e., the floor slabs). Horizontal displacements of the controlled DoFs were solved for a prototype time step using the explicit Newmark time integration method. These displacements were then computed and applied on the structure by horizontal actuators at each storey with a laboratory time step of 0.002 s corresponding to the sampling of the controllers. The forces measured by the load cells in the actuators, following the application of the controlled displacements, represent the restoring forces that are fed back to the computer and that are used in the next time step of the calculation. Restoring forces are
thus obtained from the specimen's response and reflect its state of damage. Since the inertial and viscous damping forces are modelled in the computer, the test does not have to run in real time scale. Hysteretic damping is automatically accounted for through inelastic deformation and damage progression of the test structure; consequently no viscous damping matrix was used (Molina et al. 2011). During the PsD test campaign the equation of motion was solved for restoring forces coming from the North frame only (calculated from static equilibrium of the load cell force measurements at each floor) and multiplied by a factor of two (the South frame is considered equal to the North frame in the numerical model). Equal displacements were applied to the two frames, in order to maintain zero rotation along the horizontal plane of the floor.

### 3 EXPERIMENTAL RESULTS AND SHORT DISCUSSION

Displacement transducers were installed to measure local displacements in critical areas. In particular, transducers were placed to monitor slip and crack opening between all walls and their bounding beams and columns, the displacements between the ground floor walls and the foundation beams, and the shear deformations of the two ground floor walls. Displacement transducers were also installed to measure at all storeys the vertical elongation of the bounding columns.

Inclinometers were used to measure the rotation of beams and columns at the first floor. They were placed at the centre joints and on beams and columns 30 cm away from the joints. Inclinometers were also placed at selected columns 30 cm above the foundation beam.

Heidenhain linear encoders were installed on two reference frames to measure the horizontal displacement of the two frames at each of the four floors in the direction of testing.

For the low-level acceleration, the structure behaved very well. There were no visible cracks on either the columns or the walls. Some hairline cracks that appeared on the surface of the wall closed down when the experiment was finished. The maximum top storey displacement was 24 mm and the displacement in the opposite direction was 26 mm. These displacements were the same for both the north and south frames, since the forces in the south frame were imposed in such a way so as to keep the displacements of the two frames equal, and hence avoid any torsional effects on the specimen. It was considered that both walls have reached their cracking moment, and this was the purpose of this experiment.

The 0.25g test was performed the day after the first test, and it was designed to bring the specimen at its ultimate capacity. The maximum top storey displacement was 109 mm and the displacement in the opposite direction was -93 mm. Some difference was observed in the base shear between the two frames. The maximum base shear in the positive direction (specimen was moving towards the East, i.e. towards the reaction wall) was 1074 kN for the south frame and 1036 kN for the north, which are about the same, while a negative base shear of -843 kN was recorded for the south frame and -1011 kN for the north one in the negative direction (specimen was moving towards the West, i.e. away from the reaction wall). This was an indication that the south frame has suffered some damage and it could not take further load. The only visual indication of this was a crack that opened in the ground beam at the base of the wall and the lap-splice failure due to tensile forces of the outer column on the east side of the south frame. It should be noted that the presence of the CFRP on the bounding columns of the wall, have prevented a similar failure and it allowed the completion of the experiment.

Regarding general behaviour of the specimen, it can be said that it withstood the loading imposed on it very well. There were no visible diagonal cracks on the walls, which behaved in a flexural mode. In nearly all the columns, a horizontal crack appeared at a height of 0.55 m where the lap-splice stopped, and in some cases, there was spalling of the concrete cover. Some vertical cracks appeared at the connection between the beams and the columns, but there was no severe damage, despite the fact that there were no ductile connections in the structure. In general, the stronger north frame had an overall better behaviour compared to the south one, but the differences were minor.

Finally, in the “funeral” cyclic test a displacement-history was imposed at the top storey (92, -92, 89, -125, 37, 0 mm) and a triangular distribution of forces was imposed. The objective of the test was to obtain a 20% reduction of the peak strength of the infill, in order to establish the strength envelope
of the specimen. The base shear versus the top storey displacement is shown in Fig. 4. As it can be observed in the first cycle, the structure could reach 92 mm in both directions. In the second cycle, the attempt was to reach 125 mm, but in the positive direction only 89 mm was possible to be reached, while in the negative direction 125 mm was reached. This though had as a result a sudden drop of the strength of the south frame from 838kN at -110 mm to 553kN at a displacement of -125 mm. This amounts to a drop in strength of 34%. After that, the displacement was reduced to 37 mm in the positive direction and from there to zero. Detailed presentation of the experimental model and testing campaign, as well as discussion of the results are given in Chrysostomou et al. (2013 and 2014) and Poljansek et al (2014).

The first page must contain the Title, Author(s), Affiliation(s), Keywords and the Abstract. The second page must begin with the Introduction. The first line of the title is located 3 cm from the top of the printing box.

<figure><img src="image_5.png" /><figcaption>Figure 4. Base shear versus top storey displacement for the south and north wall for the cyclic test</figcaption></figure>

## 4 2D FINITE ELEMENT MODEL

As already mentioned, during the experiments equal displacements were applied to the two frames in order to maintain a zero rotation along the horizontal plane of the floor. The loading being unidirectional, a 2D finite element model was first chosen. All the calculations presented in this section have been performed before the experiments (“blind” calculations). Only one of the two frames was considered of width 0.25m, with 75000kg per floor and a total weight of 1211kN at its base. The finite element code Cast3M developed at CEA in France was used for the calculations (Millard A., 1993). Concrete was modelled using classical quadrilateral finite elements (Fig. 5(a)) and horizontal and vertical reinforcement bars with truss elements (Fig. 5(b)). Perfect bonding was assumed between concrete and steel, the existing structure and the RC infilling. Table 1 provides the distribution of the reinforcement bars in the finite element model. The FRP's were not considered.

<figure><img src="image_6.png" /><figcaption>Figure 5. 2D finite element discretisation, (a) concrete mesh, and (b) steel mesh</figcaption></figure>

<table><tr><td>Beam (long.)</td><td>4 Φ12 (up), 4 Φ12 (down)</td></tr><tr><td>Beam (transv.)</td><td>Φ8/200</td></tr><tr><td>Column (transv.)</td><td>2 Φ20 (left), 2 Φ20 (right)</td></tr></table>
<table><caption>Table 1. 2D finite element model: distribution of reinforcement bars</caption><tr><td>Column (transv.)</td><td>Φ8/200</td></tr><tr><td>Wall (long.)</td><td>Φ12/200 (2 layers)</td></tr><tr><td>Wall (transv.)</td><td>Φ12/200 (2 layers)</td></tr></table>

Results of the modal analysis are provided in Fig. 6. The first mode, corresponding to in-plane flexure and dominates the behaviour of the structure.

<figure><img src="image_7.png" /><figcaption>Fig. 6 2D finite element model: the first four eigenmodes and the corresponding eigenfrequencies ((a) f1= 2.85 Hz, (b) f2= 11.35 Hz, (c) f3= 13.6 Hz, and (d) f1= 15.3 Hz)</figcaption></figure>

Advanced constitutive laws, suitable for cyclic loading were used for the materials. Concrete behaviour was reproduced using a damage mechanics law that takes into account the difference in tension and compression and the unilateral effect (opening and closing of cracks, Faria et al. 1998). Steel was modelled using a classical plasticity law able to reproduce the Bauschinger effect (cinematic hardening, Menegoto M., and Pinto P., 1973). The material parameters are shown in Table 2.

<table><caption>Table 2. 2D finite element model: material parameters</caption><thead><tr><th>Concrete</th><th>Parameters</th><th>Steel</th><th>Parameters</th></tr></thead><tbody><tr><td>Young's modulus</td><td>20E+09 Pa</td><td>Young's modulus</td><td>200E+09 Pa</td></tr><tr><td>Poisson coefficient</td><td>0.2</td><td>Poisson coefficient</td><td>0.3</td></tr><tr><td>Tension stress limit</td><td>3.6 MPa</td><td>Stress limit</td><td>460 MPa</td></tr><tr><td>Tension deformation limit</td><td>1.8E-04</td><td>Deformation limit</td><td>0.0023</td></tr><tr><td>Compression stress limit</td><td>33.8 MPa</td><td>Stress ultimate limit</td><td>515 MPa</td></tr><tr><td>Compression deformation limit</td><td>-3.0E-03</td><td>Deformation ultimate limit</td><td>~ 0.25</td></tr></tbody></table>

Nonlinear calculations were performed considering a very low level of viscous Rayleigh damping (0.25%), so as to be compatible with the PsD testing (see section 2). The implicit Newmark numerical algorithm was adopted for the time integration of the equation of motion and the classical Newton-Raphson algorithm for the nonlinear material behaviour. Results are presented hereafter for 0.25g (and for an earthquake signal of about 12 sec).

Figure 7 compares the numerical and the experimental results considering the displacements time histories at all floors. The performance of the “blind” numerical model is satisfactory in terms of maximum values and frequency content especially for the upper floors.
<figure><img src="image_8.png" /><figcaption>Fig. 7 2D finite element model (0.25g): displacements time history, experimental vs. numerical results</figcaption></figure>

Figures 8 and 9 show the performance of the model in terms of base shear-top displacement and storey shear-interstorey drift at each floor. Results are particularly good for the upper floors. It can be noticed, however, that the model does not reproduce accurately the interstorey drift at the base.

<figure><img src="image_9.png" /><figcaption>Fig. 8 2D finite element model (0.25g): Base shear - top displacement, experimental vs. numerical results</figcaption></figure>

<figure><img src="image_10.png" /></figure>
Fig. 9 2D finite element model (0.25g): Storey shear and interstorey drift, experimental vs. numerical results

The advantage when using continuum mechanics constitutive laws is that we have access to local
results in terms of damage indices, strains in the reinforcement bars etc. Figure 10 shows the damage
index due to tension (a) and to compression (b) for a loading of 0.25g. This damage index varies be-
tween zero and one (from undamaged to completely damaged cross-section). Therefore, a zero damage
index corresponds to no tension or compression, whereas when the variable reaches the value of one it
corresponds to the attainment of the tension or compression capacity of the members. It is clear from
Fig. 10 that the structure has exhibited good behaviour in terms of the strong column-weak beam
mechanism. However, the existence of damage due to compression at the bottom (Fig. 10b) indicates
that this level of loading is important and that for a higher level of PGA the structure maybe at risk due
to possible compression failure of the columns bounding the wall.

<figure><img src="image_11.png" /></figure>

5 SIMPLE NUMERICAL MODEL

The analysis of the building was also conducted using a simple numerical model based on line and shell elements. The purpose of this model was to simulate the response of the specimen as it was tested in the lab, unlike the 2D micro-model presented in the previous section by which only one of the two frames was modelled, observe the correlation of the numerical to the experimental results, and examine the effect of modelling in increasing the accuracy of simulation. The simplicity in modelling results in computational-time reduction and thus provides the opportunity to study the 3D behaviour of the building. It should be repeated here that the PsD test was unidirectional and care was taken so no rotation was introduced in the two frames. Therefore, by 3D behaviour it is meant the contribution of the connecting slabs and beams to the response of the two planar frames. The numerical model was created in SAP2000 (2010), which is widely used both for commercial and academic purposes (unlike the sophisticated software needed for micro-modelling), offers a friendly graphical interface, and includes non-linear models both for line and shell elements. A view of the model is shown in Fig. 11.
<figure><img src="image_12.png" /><figcaption>Fig. 11 3D view of the numerical model in SAP2000</figcaption></figure>

The modelling of the building was assembled using line fibre elements for the columns, which are expected to behave nonlinearly as it was observed during the experiment. The column cross-section was discretised in eight concrete fibres and four steel ones as shown in Fig. 12. The fibre element properties were distributed along the length of the column elements.

The Takeda (1970) model was used to describe the nonlinear properties of concrete including cracking. The Mander (1984) unconfined concrete model was used to describe the monotonic stress-strain relationship. In the case of the steel reinforcement, kinematic hysteretic type was assumed. The monotonic laws for both materials are shown in Fig. 13.

<figure><img src="image_13.png" /><figcaption>Fig. 12 Characteristics of fibres of column cross-section</figcaption></figure>

Both longitudinal and crossbeams were modelled using elastic properties since they did not show nonlinear behaviour during the experiments. In order to account for the reduction in stiffness due to cracking at the 0.25g test, a 10% reduced elastic stiffness was used for the longitudinal beams. The slabs were modelled with linear shell elements.

<figure><img src="image_14.png" /><figcaption>(a) Concrete</figcaption></figure>

<figure><img src="image_15.png" /><figcaption>(b) Steel reinforcement</figcaption></figure>
<caption>Fig. 13 Monotonic laws</caption>

RC infill walls in the middle bay of the building were modelled using a nonlinear fibre shell element. The discretization of the wall cross-section was created automatically and accounting for the in-plane properties of the wall. The steel mesh reinforcement of each wall was modelled based on the corresponding detail as described in Chrysostomou (2014) in order to account for the strength reduction in higher floors. The discretization for the first floor wall is shown in Fig. 14. The same material constitutive laws were applied as for the column line elements. A single shell element was used to model the wall in each floor. The walls were assumed to be rigidly connected to the four corners of the bounding frame elements in each floor therefore no relative movement between the walls and the bounding frame was allowed, which is one of the shortcomings of the model.

Based on the correlation of the results, it will be concluded whether such a movement took place or if the connection of the wall to the bounding frame using closely spaced dowels was close to rigid.

<figure><img src="image_16.png" /><figcaption>Fig. 14 Discretization of shell RC wall cross-section for the 1st floor wall</figcaption></figure>

The full mass of the original building (600 Tonnes) was applied on the model as lumped masses on all the nodes. The floor loading included floor finishes and distributed masonry walls. The numerical period at the beginning of the 0.25g test was equal to 0.48sec., which was very close to the experimentally recorded one at the beginning of the test that was approximately equal to 0.50 sec. The evolution of the first mode frequency during the 0.25g test is shown in Fig. 15.

<figure><img src="image_17.png" /><figcaption>Fig. 15 Evolution of frequency of the tested building during the 0.25g test</figcaption></figure>
Similar to the 2D finite element model, a 0.25% viscous damping was used in the model to account for the limited viscous damping activated in the structural elements during the slow process of the pseudo-dynamic testing. The hysteretic damping introduced by the nonlinear behaviour of the elements was modelled by the nonlinear hysteretic models used for concrete and steel.

The comparison of experimental to the numerical history of top storey displacements is shown in Fig. 16. It can be easily concluded that the numerical model captures with high accuracy the behaviour of the tested building. Both the evolution in period and strength during the experiment are very well simulated in the model and so is the nonlinear behaviour of the building.

<figure><img src="image_18.png" /><figcaption>Fig. 16 Comparison of top storey displacement history for the 0.25g test</figcaption></figure>

To examine the correlation of the base shear force, energy dissipation and stiffness between the experimental and numerical model, the base shear versus top displacement hysteretic curves were plotted as shown in Fig. 17.

<figure><img src="image_19.png" /><figcaption>Fig. 17 Comparison of base shear vs. top storey displacement history for the 0.25g test</figcaption></figure>

From the comparison in Fig. 18, it appears that the base shear in the numerical model simulates closely the experimental one. In addition, the stiffness of the numerical model defined by the slope of the hysteretic loop is close to the experimental one. Regarding the energy dissipation, based on the width of the hysteretic loops it appears that the numerical model absorbs slightly higher levels of energy (probably more damage).
# 6 DERIVATION OF FRAGILITY CURVES

Fragility curves are regarded as a viable tool for seismic risk assessment of existing buildings. Seismic fragility is defined as the damage state probability conditioned on a seismic hazard parameter. In structural assessment terms, the fragility of a structural system is defined as the probability of reaching a particular structural-related damage level defined using structural response parameters conditioned at a specific seismic hazard. The seismic hazard parameter can be expressed in terms of Peak Ground Acceleration (PGA), velocity/displacement or in terms of spectral values of the above mentioned quantities. The most common seismic hazard parameter used in the literature is PGA since it is used to quantify the seismic hazard from pertinent maps that are introduced in modern design codes. In addition, instrumental acceleration recordings through accelerometer networks are continuously gathered in most seismically-active areas providing information regarding the spatial distribution of the PGA level. After amplification through simple operations (e.g. the design spectrum equations) the PGA is used to estimate the response acceleration, which in turn is related to the seismic inertial force applied on the structural system.

In order to establish the fragility relationship between the probability of reaching or exceeding a particular damage state with a commensurate increase in the PGA value, the definition of the damage state needs to be articulated. In the case of older non-seismically designed RC buildings retrofitted with RC Infill walls such curves do not exist in the literature. Their derivation relies strongly on the capacity of the RC wall and the selected damage states should follow the damage history on the RC wall. As it was seen from the experimental results, damage is concentrated on the wall at the base, which exhibits yielding and ductile behaviour. Based on the recorded results from the pseudo-dynamic test, that are described in detail in the preceding sections of the paper, evidence of damage in the wall appears at the end of the first test in the form of cracking (minor damage). This damage though is not significant enough to be accounted for in a fragility assessment framework. The second pseudo-dynamic test at PGA=0.25g pushed the building at approximately 100mm top displacement causing severe damage to the RC Wall at the base due to flexural cracking. Based on the LVDT recordings the wall has reached and exceeded its flexural capacity, which, given the shape of the base shear vs. Top displacement curve, was reached at approximately 65mm. By the end of the test, the 100mm top storey displacement can be regarded as a near collapse damage state based on the observed damage. The third and final test (cyclic test) pushed the building to a top displacement of 125mm, which caused severe drop of its strength capacity. This drop can be regarded as an indication of collapse initiation through instability.

Through the above mentioned recordings and observations it is possible to define four damage states for (a) cracking of the RC Wall, (b) initiation of yielding in the reinforcement of the RC wall, (c) severe structural damage at the base of the RC wall and cracking at the interface with the foundation, and (d) collapse of the building due to instability.

The top displacement value for the four damage states listed above was identified from the experimental results and is used herein for the derivation of fragility curves. Since this values were obtain through a deterministic process (test), they are treated as the mean value of the logarithmic distribution describing the fragility curve of each damage state. It should be noted though that a large variability is associated to the mean value due to both capacity and demand uncertainties. This variability related to this mean value is expected to arise from three primary sources (FEMA-NIBS 2003):

* variability in the seismic hazard related to signal variations (as far as duration, frequency, number of peak values etc.)- thus, various sources of variability of the demand imposed on the structure through the earthquake ground motion
* variability in the strength and deformation capacity of the frame elements (variability associated with the corresponding capacity of each structural response type). This is related to material properties (strength and strain capacities), detailing properties etc.

* variability in the derivation of the analytical capacity models derived in Pardalopoulos (2013), which is associated with the discrete threshold of each damage state. This is expected to vary depending on the failure mode considered and might even be zero in cases where first principles were used to derive the models.

<caption>Figure XX. Fragility curves for the selected damage grades</caption>

In order to accommodate the first two uncertainties in the derivation of the fragility curves, the work from Kyriakides (2015) on private buildings in Cyprus was examined to obtain the standard deviation values associated with the derived fragility curves. In the work, deviations in material properties were examined in detail. Several non-linear time history analyses were conducted to obtain the structural response of both old and new RC frame buildings in Cyprus. It was concluded from the study that for mid-rise non-ductile RC frames the standard deviation from the variability in seismic hazard and capacity of the frame elements adds up to σ=0.6. This is in line with the results of a similar study conducted by Kappos et al. (2008) for similar construction practice in Greece and is used herein as an estimate of the capacity and demand uncertainty.

Assuming a lognormal distribution (common assumption in seismic fragility studies (e.g., Anagnos et al. 1995), the conditional probability of being in or exceeding, a particular damage state dsi, given the Sd is defined by the relationship,

$$P[ds \geq \frac{ds_i}{PGA}] = \Phi \left[ \frac{\zeta}{\beta_{dst}} \ln \left( \frac{PGA}{PGA, ds_i} \right) \right]$$

where:

Sd, dsi is the median value of peak ground acceleration at which the building reaches the threshold of damage state, dsi

βdsi is the standard deviation of the natural logarithm of PGA for damage state, dsi, and

Φ is the standard normal cumulative distribution function.

Each fragility curve is defined by a median value of Sd that corresponds to the threshold of that damage state and by the variability associated with that damage state. This value is obtained deterministically from the pseudo-dynamic test results as described in previous paragraphs. The Sd calculated as described in the framework is treated as the median value and lognormal standard deviation values (β) describing the total variability associated with each fragility curve are attributed accordingly based on results from Kyriakides (2015). The derived fragility curves are shown in the figure below.
<figure><img src="image_20.png" /><figcaption>Figure XX. Fragility curves for the selected damage grades</figcaption></figure>

# 7 CONCLUSIONS

The purpose of this paper was to examine the capabilities of a 2D finite element micro-model and a simple fibre macro-model to simulate the experimental nonlinear behaviour of a 4-storey RC building strengthened with RC infill walls and use the latter and more computationally simple one for the derivation of fragility curves. The numerical results from the two models were compared in the paper with the experimental results from a full-scale pseudo-dynamic test on a retrofitted RC building. The comparison was conducted using the floor displacement and base-shear time-histories. The elastic characteristics of the building were also well simulated as it is shown from the comparison of the fundamental frequency values. Both models managed to simulate the experimental behaviour of the tested retrofitted building with very good levels of accuracy in all the response parameters mentioned above. Since no relative movement was allowed in the interface between the RC Infill walls and the bounding frame in the numerical models, it was concluded that using the specific connection details leads to monolithic behaviour. The procedure followed for the design of the connections is described in Fardis (2013). It should be stressed out though that if less dowels were used at the connection interface, relative movement might have taken place, which would require modelling of the interface between the infill and the bounding frame in order to capture this behaviour, hence more complicated models would have been required. The recordings and damage observations during the experiment were used to define viable damage states representative of the damage history during the three tests. Four damage states were defined ranging from cracking of the RC wall to collapse of the building due to instability. The corresponding recorded top storey displacements for each damage state were used as median values in a logarithmic distribution model to derive the relationship of probability of reaching or exceeding a particular damage state for a given spectral displacement (Sd) value. Uncertainty in the curves was incorporated based on literature. It is clear from the fragility curves that a considerable amount of ductility (μ~2) is achieved prior to the collapse of the building due to the insertion of the RC Infill walls and this is regarded as a success of the retrofit method.

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