Global Interventions for Seismic Upgrading of Substandard RC Buildings

: A methodology for design and proportioning of interventions for seismic upgrading of substandard reinforced concrete (RC) buildings is developed in this paper. The retrofit approach is presented in the form of a simple design tool that aims at both demand reduction and enhancement of force and deformation supply through controlled modification of stiffness along the height of the building. This objective is achieved by engineering the translational mode-shape of the structure so as to optimize distribution of interstorey drift. Results from the proposed approach are summarized in a spectrum format, where demand expressed in terms of interstorey drift is related to stiffness. Design charts, which relate the characteristics of commonly-used global intervention procedures to influence drift demands are developed to facilitate the retrofit design. Intervention procedures considered in this paper are reinforced-concrete jacketing, addition of reinforced concrete walls, and addition of masonry infills. The proposed methodology is also amenable to adaptation to other strengthening methods such as addition of cross-bracing and others.


INTRODUCTION
Earthquakes have repeatedly illustrated the deficiencies of older reinforced concrete construction built prior to the proliferation of contemporary earthquake design principles. A broad category under this description represents structures with vertical irregularities in distribution of stiffness or mass, such as multistory residential buildings with open first storey, which are known to be susceptible to soft-storey formations leading to severe damage or collapse (Fig. 1).
Under earthquake loading, failure will occur where deformation demand exceeds the deformation capacity of the gravity load-carrying members of the structure (i.e.. in columns).
Therefore, in order to protect vulnerable structures, an effective approach is to enhance the capacity and simultaneously reduce the demand. In the context of the present paper, deformation demand is quantified by interstorey drift (ID) throughout the structure. Therefore, ID demand depends entirely on the global amount and distribution of structural stiffness at least up to the point of yielding in conventional pushover analysis. Deformation supply refers to the individual elements of the structure; it is controlled by availability of confinement, shear reinforcement, lap and anchorage conditions, ratio of compression to tension reinforcement and amount of axial compression. Many (but not all) of those deficiencies in deformation capacity can be alleviated through local interventions.
In the context of a rehabilitation strategy framework, the proposed methodology aims at systematic reduction of deformation demand, and in particular, elimination of any tendency for localization of demand in parts of the structural system. The degree of stiffness irregularity in the structure and the resulting local increase in the magnitude of demand (i.e., the magnitude of imposed ID) during an earthquake may be diagnosed by the morphology of the fundamental translational mode of vibration. For example, a uniform distribution of ID would correspond to a linear first mode shape; in this case, the lateral translation increment in the mode shape coordinate from floor to floor equals to 1/n, where n is the number of floors ( Fig. 2(a)). A shear-type first mode is marked by higher increments in the lower floors gradually decreasing towards the upper floors ( Fig. 2(b)). The reverse pattern occurs in a flexural-type translational mode (Fig. 2(c)). In a soft storey formation, the mode shape is practically constant above the soft storey (that is, negligible ID occurs in all floors but within the soft storey, Fig. 2(d)).
In a reverse process of redesign, where the desirable pattern of ID distribution prescribes the proper morphology of the fundamental mode shape, it is relatively straightforward to evaluate the pattern of stiffness distribution throughout the structure, required to produce a desirable translational mode. The necessary stiffness that is estimated from this process can be added through pertinent interventions in each floor as required.
Dimensioning and detailing of these interventions refers to basic mechanics of reinforced concrete. Examples of stiffness-modifying interventions are, (a) reinforced concrete jacketing of columns, (b) addition of reinforced concrete walls, (c) addition of masonry infills or steel cross-braces, (d) addition of metal or FRP longitudinal reinforcement in columns, properly anchored in the ends, and combined with pertinent transverse jacketing for confinement. Note that implementation of some of these interventions (e.g., case (b)), may require a more extensive solution that would also involve upgrading of the foundation, whereas other methods (such as for example, case (d)) are only supported so far by very limited experimental evidence.
In the remainder of this paper, design calculations, including stiffness values, refer to the onset of yielding (secant stiffness values). Thus, vertical members are dimensioned to match the required stiffness and a selected target value of ID y,target at yielding. The process is facilitated by development of the Interstorey Drift Spectra (IDS), which relate the characteristics of familiar global intervention procedures to the magnitude of interstorey drift demands (IDD) during earthquakes. These results may also serve as rapid inspection tools that facilitate immediate assessment of the effect on IDD, produced by changes in the technological details of the intervention methods.

Controlled distribution of damage along the height of the building
IDD is considered in earthquake engineering as the most representative index of damage assessment, and for this reason it is related directly to definition of performance objectives both in design and rehabilitation. For a given level of building lateral translation, limiting the magnitude and distribution of IDD in the structure may be managed by controlling the relative displacement pattern implicit in the modes of vibration that mostly affect dynamic response. In the proposed rehabilitation framework, the lateral response of the building is explicitly modified through retrofitting, by targeting towards a desirable pattern for the fundamental mode of translational vibration.

Selection of the appropriate target response shape
A point of reference in selecting the target response shape are, the fundamental mode and period of the existing building. These may be evaluated rapidly through standard Rayleightype or Stodola-type iteration (Clough and Penzien 1993), using secant-to-yield stiffness values for the individual members. The estimated fundamental translational response shape and period may guide definition of retrofit objectives: an excessively large fundamental period value identifies the need of lateral strengthening of the building (owing to the implicit relationship between stiffness and strength) by means of controlled stiffness addition along the building height. Proportioning the stiffness of the individual floors is determined so as to even out large discrepancies in relative drift between successive floors detected in the fundamental response shape pattern.
For lightly reinforced frames or flat-slab structures an acceptable retrofit scenario may be developed by targeting at a response shape that ranges between a controlled shear-and the triangular profile (Fig. 2). Complete alteration of the fundamental response shape from a shear to a flexural -type profile would require addition of excessive amounts of lateral stiffness to the structure. For usual pilotis buildings ( Fig. 1, 2(d)) this would most likely exceed the strength and stiffness needed in order to achieve a commonly acceptable frame response.

Achievement of target deformation shape by stiffness adjustment
Consider a planar frame structure with lumped floor masses vibrating in a single lateral displacement shape, Φ. If the shape is normalized with respect to the displacement at the top of the building, ∆ top , it follows that the i-th floor displacement is, ∆ i =Φ i ∆ top and the corresponding interstorey drift is, The generalized properties of the equivalent single degree of freedom system (ESDOF) and its associated equation of motion are given by (Clough and Penzien 1993): where L * =Σm i Φ i , whereas, m i and K i are the translational mass and stiffness terms of the i-th floor (K i corresponds to onset of yielding). Note that contribution of a floor's translational stiffness to the generalized structural stiffness K * is controlled through the term ∆Φ i in (Eq. 1).
Corollary to this is that a floor does not contribute to the generalized stiffness if the corresponding floor value of ∆Φ i is zero -this is the extreme case of structures with a soft first storey ( Fig. 2(d)) where ∆Φ i ≈0 for the upper floors (i.e., although these floors have larger stiffness than the soft storey, they do not contribute to the generalized translational stiffness of the structure).
The reverse problem of establishing the required translational stiffness values in order to achieve a target shape Φ i , is solved from the eigenvalue problem of a planar multistory structure with lumped properties and translational degrees of freedom (d.o.f.): where m and F are the mass and flexibility matrices of the structure. By definition F ij is the displacement at d.o.fj, produced by a unit force acting along the d.o.fi. For frame structures with one translational degree of freedom per individual floor (i.e., when floor rotations are either negligible or condensed out of the equation of motion in the absence of associated mass), the above takes the following form: Terms K 1 , K 2 , etc. are the work-equivalent translational stiffnesses of the individual floors (degrees of freedom are numbered from the first storey upwards). For a given target shape (i.e., for known Φ i values), the first equation above is solved for the required value of K 1 , next, the second equation is solved for K 2 , and so on. Clearly the system with stiffness distribution satisfying Eq.
(2) will have a fundamental response shape equal to the enforced target shape, Φ. Note that since ω is a multiplier, the above equations prescribe the required ratio between various floor stiffness values (κ 2 =K 2 :K 1 , κ 3 =K 3 :K 1 ,…, κ n =K n :K 1 ) in order for the building to achieve the target shape. For equal mass, Fig. 3  ton. For other mass values, the ordinate axis must be multiplied by this value).

Storey stiffness contribution to generalized stiffness
The translational stiffness of the i-th storey, termed as K i (i=1, 2, … n) in Eq. (2b), and its contribution to the structure's generalized stiffness K * in Eq. (1), is given by the sum of the work equivalent stiffness terms of its individual members (j=1,2,…ℓ). To calculate the work equivalent stiffness contribution of the j-th element (beam, column, or wall) of the i-th floor, consider the chord rotations θ j 1 , θ j 2 developing in the ends of that member when the structure deflects laterally following the applied deflection shape, Φ ( Fig. 5(a)); end moments of each member are obtained from the deformational member stiffness matrix: Coefficient ß accounts for the contribution of shear deformations in the response; for frame members, ß j may be taken equal to zero without great loss of accuracy (i.e. neglecting shear deformations), whereas in the case of walls, terms (4+ß)/(1+ß) and (2-ß)/(1+ß) account for the work done by shear deformations in the wall in addition to that by flexural curvature.
Term ß=24a sh (1+ν P )(r w /h i ) 2 , where r w is the radius of gyration of the wall cross section in the direction of main action, a sh is the ratio of the walls's shear area to its total cross section (≈0.8), and ν P is the material's Poisson's ratio (α sh (1+ν p )≈1). It is also assumed that the deformed structure is forced to undergo a virtual displacement pattern which, for convenience, is taken identical to the deflected shape, i.e., δΦ=Φ ; based on this postulate, virtual rotations δθ j 1 , and δθ j 2 at individual member ends are respectively equal to θ j 1 , θ j 2 . The resulting virtual work expression for a single storey is: The notation used {…} i denotes that the operation inside the {} concerns the ℓ- δθ j 1 = θ j 1 ; δθ j 2 =θ j 2 ).
Wall systems or wall-equivalent dual systems: Storey stiffness contribution as described by Eq. (4) is written in general form as: The moment equilibrium at the beam-column joints depends on the relative stiffness ratio of the n b beams and the n c columns that converge at a typical floor joint:  Column deformation (term λ c θ 1 of Eq. (7)), is different from interstorey drift. Here it is recognized that a part of the interstorey drift is owing to rotations occurring in the beams, referred hereon as "tangential" interstorey drift and established from the derivative of the deflected shape at the top of the column in consideration.
Considering that the typical frame structure responds following a shear or at most, a triangular response profile and that column length, h c , is equal to storey height, h i , for a representative average value of λ for the entire storey, the general expression for the i-th floor stiffness as defined by Eq. (6) is further modified to: The value of λ used in Eqs. (6), (7) and (8) is the average "rotational stiffness ratio" value, calculated for all beam-column joints that belong to the floor in consideration.
Coefficient α c,b =α c,J =12 refers to full end restraint against rotation (θ j corresponds to release of rotational restraint at one end, whereas intermediate values correspond to partial rotational restraints at the member ends. In the triangular response shape the lateral drift is accommodated by deformation at the lower end of the first storey columns only, θ c,j 1 =1/h i , θ c,j 2 =0, and at all the beam ends, θ b,j 1 =θ b,j 2 =1/h i (in adopting a triangular response shape it is implicitly assumed that yielding occurs at the bottom of the first storey columns and at the beam ends. Note that in this case α c,b =12 in Eq. (8), whereas α c,J =4 for i=1 in Eq. (8), and α c,J =0 for i>1).
In the postcracking range of response, if a strong-column/weak-beam design is implemented in the retrofit, the flexural stiffness ratio follows that of the moment strength ratio (<1:1.4); assuming that the beam span to storey height ratio L b,j /h c,j is near the value of 2 (bending in double curvature), it follows that a value of λ=1/(2x1.4)=0.36 is representative for interior connections (n b =n c =2) (more generally λ<1), leading to λ c ≈0.24 and λ b ≈0.76.

Proportioning of retrofit for target shape upgrading
In order to practically implement the required storey stiffness K i of the retrofitted structure the stiffness of those members that participate in the global intervention scheme is expressed in terms of the technological details of the retrofit.
(1) Reinforced concrete walls: Addition of either new RC walls or infill walls (partial or full) in strategically-selected bays of the existing frame is also a common method used for strengthening of existing structures. This is particularly suitable for structures with a poor frame action (flat-slab structures) that suffer from an inherent deficiency of lateral stiffness. If the wall occupies a full bay, then it is often designed to incorporate the beams and the two end-columns of the bay, the latter acting as its boundary elements. This method efficiently controls global lateral drift, thus reducing demand in vertical frame members. A prime consideration in design is distribution of the walls so as to avoid plan eccentricity. Other issues concern provisions to secure safe transfer of inertial forces to the walls through floor diaphragms, struts and collectors, integration and connection of the wall into the existing frame buildings and transfer of loads to the foundation. Added walls are typically designed and detailed following current code requirements for new structures. A typical wall cross section is depicted in Fig. 6(a). The relationship between the normalized compression zone depth ξ w (=c/d w ) and the longitudinal reinforcement ratio of the boundary columns, ρ be , is plotted in Fig. 6 Thus, Eq. (5) may be written as follows: (2) Reinforced concrete jacketing is the most common rehabilitation method for concrete buildings in Southern Europe. Apart from enhancing the deformation capacity of the retrofitted columns, an advantage of jacketing is that it may achieve a more uniform distribution of stiffness and strength throughout the plan of the building as compared with the addition of shear walls, while avoiding the pitfalls of the latter approach which invariably requires extensive redesign of the building's foundation in accord with capacity design considerations. Another advantage is that column jacketing continues through the floor diaphragms, thereby encasing the regions of beam-column joints, eliminating the risk of a joint shear failure in the retrofitted structure. A practical difficulty is caused by beams at the beam-column connections, which may require bundling of added longitudinal bars in the corners of the column jacket.
A typical jacketed cross section is idealized in Fig. 7(a) having initial dimensions b c and h c , increased to b J , h J after jacketing. The compression zone depth c, is normalized as ξ J =c/d J ; its relationship to the total equivalent reinforcement ratio, ρ e , and axial load ratio, ν , is plotted in Fig. 7 Similarly, the secant-to-yield cross-sectional stiffness of a RC beam, E c I b y , and associated translational floor stiffness of the frame structure (from Eq. 8) is: (3) Infill masonry walls: Adding infills as a means of retrofitting moment resisting frames (MRF) is a popular method in Southern Europe and it is encouraged by EC8-III (2005); in North America this retrofit method is considered controversial, as it is thought to be increasing the mass of the system without any effect on strength at large ductilities -a counterargument is that by adding stiffness in the range of elastic frame response, displacement demands are moderated, provided the wall is connected through its thickness.
Caution should be exercised should this option be pursued; the translational stiffness of an infill masonry wall deforming in its plane is estimated with reference to a diagonal strut used to idealize the infills' function as a stiffening link (EC8-I 2004, EC8-III 2005 Secant stiffness of wall infills decays with imposed drift demand. Note that infills "yield" at drift levels in the range of 0.15 %-0.2% ; thus, when the surrounding frame elements are at yielding, i.e. at an interstorey drift in the range of 0.5%, the wall is already at a ductility level of 3-4.
Any contribution of infill wall stiffness to out-of-plane action is neglected.

Relationship between stiffness and floor area ratios of lateral load bearing members
Equations (9), (11) and (12) relate sectional properties to member detailing; thus, translational member stiffnesses given by Eqs. (10) and (13) (Gülkan and Sozen, 1996): which is simplified further to: Beam contribution to stiffness in flexural-type wall structures is neglected in the remainder, for simplicity. For masonry infills: Terms Ω b , Ω J,c , Ω w and Ω mw control the stiffness of a particular floor and are parametrically related to the characteristics of the global intervention considered and the materials used. For simplicity in preliminary retrofit design, and to limit the degrees of freedom in decision making, it is possible to initially neglect the influence of beams, by setting the corresponding λ parameters in Eq. (17a) equal to: The corresponding elastic value of the i th storey interstorey-drift, ID i , is given by: For any chosen target response shape, values of L * , M * calculated from Eq. (1) are substituted in Eq. (19) to extract expressions for the elastic IDS. In frame structures, the column share of the elastic interstorey drift is ID i c =λ c ·ID i , whereas the corresponding beam Assuming that the critical storey is the first storey (ground floor, with a typical storey height h i =3 m) the general form for the elastic interstorey drift of the first storey, ID 1 , is (Units in ton, kN, m): where Ω 1 is the stiffness term of the first storey given by the sum of terms Ω J , Ω w and Ω mw   Table   1). The target interstorey drift, ID y,target,i , is defined by dividing the elastic interstorey drift by the behavior factor, q (q=µ in the higher range of periods except for low periods). Thus, for a known number of storeys, using the target drift level of the first storey as the sole input to the retrofit design algorithm, the demand in stiffness of the first storey is obtained directly. It is observed that in case of the triangular and the shear response profiles, for drift levels equal to 1.5% or higher, and for buildings higher than three storeys, the stiffness of the first storey is insensitive to the number of storeys. Also note that for the flexural response shape, the building's aspect ratio is reflected in the value of the stiffness demand estimate of the first floor (this value is reduced after a number of floors as the building geometry becomes more slender tending naturally towards a flexural-type response).
Application of the proposed method is demonstrated in the last section of the paper through an example case study. a 1 x 1 +a 2 x 2 =C 1 , a 1 x 1 +a 2 x 2 +a 3 x 3 =C 2 where the constant term C 1 , or C 2 , depends on the drift level considered as depicted schematically in Fig. 10(a). The plots of Fig. 10(a) are referred to hereon as Interstorey Drift Spectra (IDS). They provide all possible combinations of (Ω J , Ω w , Ω mw ) values that satisfy the elastic drift level under the specified design earthquake. In practice, it is uncommon to use all three GI methods simultaneously and the 2D plot ( Fig.   10(b)) appears more applicable. The IDS of a 3-storey frame building that responds according to a shear profile is shown in Fig. 10(b). In this application, the contribution of the infill masonry walls has been neglected (Ω mw =0) and parameter λ c has been taken equal 1 to simplify the preliminary retrofit process as discussed in the preceding.
Similar IDS may be developed using the same procedures for any global intervention method, provided that the elastic stiffness may be expressed explicitly in terms of the technological parameters of that intervention (using the same steps as done herein for jackets, walls and infills.)

PRACTICAL RETROFIT DESIGN
Performance objectives are set with reference to yielding of the retrofitted structural members (e.g. ductility demand associated with the YPS used). Thus, in designing the retrofit scheme with the proposed methodology the engineer selects: (1) the target interstorey drift at yield of the first storey, ID y,target,1, (2) the target response shape, Φ, and (3) the target ductility demand, µ target (i.e. performance of the retrofitted structure is considered in the inelastic stage of response). The construction of the IDS follows (Fig. 10), where the target interstorey drift at yield of the first storey, ID y,target,1 , is related to the corresponding stiffness demand in terms of Ω J , Ω w and Ω mw drawn using the results of Eqs. (20); here, for demonstration, three alternatives for the selected target response shape are considered. Numerous combinations for the stiffness parameters are possible, depending on the target interstorey drift at yield of the first storey, ID y,target1 .
The lower the ID y,target,1 , the higher the level of required strengthening, leading to a further reduction in the period value. The demand for an ESDOF in the ADRS Spectrum format is given by the radial line with a slope of 4π 2 /T 2 , where T is the system's period. Thus, the changes effected in the period as a result of the alternative retrofit solutions may be readily inspected by this geometric parameter (i.e. the slope of the radial line).
The required floor stiffnesses Ω J,c , Ω w and Ω mw thus determined, are then distributed to the floor vertical members at ratios proportional to gross section properties. Flexible columns of the lateral load resisting system (i.e., members with a sway index>30) or columns with a high axial load ratio (ν>0.3) are primary candidates for stiffness enhancement; alternatively, all required stiffness addition may be provided by addition of RC walls or infills.
Each member is designed in the retrofit scheme to satisfy the targeted interstory drift at yield (for columns: ID y,target,1 c =λ c ·ID y,target,1 ) and to provide the required stiffness as defined through the Ω J,c , Ω w and Ω mw coefficients (Eqs. (17)). For the j-th member of the i-th floor, given the drift or rotation at yielding (ID y,i ), the corresponding curvature demand value (reinforcement slip owing to bond included), φ y,j , is calculated from φ y,j =

Verification of member ductility demand
After dimensioning and detailing, the available drift capacity may be evaluated for each retrofitted member, for the purposes of verification of the retrofit design. Assuming that all premature failure modes but flexural are suppressed through local interventions at the member level (e.g., through jacketing with composite wraps, and by capacity-designing transverse reinforcement in the RC jacketed columns), the ultimate curvature demand is expressed in terms of the member's displacement ductility demand, µ (Paulay and Priestley 1992): The solution scheme is acceptable when the ultimate curvature capacity of the member (φ u,j =ε cu /ξ u d) exceeds the estimate of Eq. (22) (ε cu is the strain capacity of the member's compression zone).

SCOPE AND LIMITATIONS OF THE PROPOSED FRAMEWORK
Relevant issues or precautions related to the scope, limitations and applicability of the proposed retrofit framework are the following:

Scope:
The proposed method is not intended to replace or substitute detailed modeling of the rehabilitated structure; rather, post-design performance verification checks should be done, whenever possible, through time history analysis of a robust structural model of the structure after the rehabilitation design has been finalized. The proposed framework is meant to address the needs of preliminary design, namely to provide a test-bed for fast assessment of alternative retrofit schemes. The method uses a spectrum-based description of the design earthquake. Therefore, it necessarily relies on an equivalent single degree of freedom representation of the structure. Other considerations also apply in determining the applicability of a SDOF-based approach as opposed to detailed Finite Element Modeling (FEM) for preliminary design, such as, budgetary limitations, lack of detailed drawings of the original old structure (a frequent occasion in many regions of the world), but also in the presence of obvious deficiencies that identify a single mechanism formation (such as soft storeys).

Limitations:
Although the emphasis is on the secant-to-yield stiffness, strength is the underlying parameter due to the implicit relationship between stiffness and strength in RC members. Thus, the reverse enforcement of a target fundamental response pattern basically regulates the strength distribution throughout the building, by exploiting the stiffness to strength relationship. This is why the analysis is conducted with reference to the yield point (end of the elastic branch) in the response curve of the rehabilitated structure. However, this does not imply that response of the rehabilitated structure to the design earthquake is to remain elastic -rather, through the use of the YPS, the choice of stiffness/strength pattern also determines immediately the targeted ductility demand, µ target , that will be imposed on the rehabilitated structure by the design earthquake. Considering that nonlinear behavior of the retrofit is often limited by the existing reinforcement anchorages, which may remain a weak zone of behavior even after rehabilitation, it is generally advisable that the ductility demand targeted for through the choice of the design YPS should not exceed the value of 3 (i.e., it should be required that ∆ u <3∆ y ) (EC8-I 2004, KANEPE 2010); this limit refers to the structural system. Having eliminated the risk of localization through proper selection of the target response shape, individual target member rotation demand is given by:

Description of the frame structure
The proposed methodology is applied to a four-storey four-bay frame designed according with the prevailing practice of Southern Europe in the 1970's (Fig. 11) Fig. 11.

Alternative retrofit options
Raleigh-type iteration of the existing frame assuming rigid diaphragms led to a period estimate of 1.66 sec and a translational mode Φ Τ =[0.27, 0.56, 0.82, 1.00] T . To reduce the period to a value near the 0.4 sec empirical limit for a four-storey frame, strengthening was considered using two alternative retrofit strategies. The first one, referred to hereon as RS1, aims at strengthening the existing frame while maintaining, albeit improving the shear-type response profile (Fig. 12(a)). The second retrofit scenario, (RS2), aims at altering the fundamental mode of the frame towards a flexural-type ( Fig. 12(b)). This is accomplished by infilling the two external bays (2 m spans) while RC jacketing the middle row of columns. In both approaches the contribution of infill masonry walls to the floor stiffness was neglected (by setting the term Ω mw =0). In the case of the RS1 retrofit solution the influence of the contribution of beams to the floor stiffness was estimated according with Eq. 17(a) through parameters λ c and λ b .

(1) RS1: Retrofit by RC column jacketing
This retrofit scenario aims at a significant period reduction and a uniform distribution of interstorey drift (i.e., triangular target response shape, Φ T =[0.25, 0.50, 0.75, 1.00] T ) by RC jacketing all the vertical members along the height of the building with a 50 mm thick jacket layer (i.e., final column dimensions were increased to 350x350mm, Fig. 12(a)). All members of the same floor participate in the lateral resistance of the structure by the same ∆Φ i value.
The EC8-I Type I (2004) earthquake design hazard was used to define demand ( Fig. 13(a)); this was expressed by the Y.P.S. obtained using the equal displacement rule (q=µ). Each radial line corresponds to a period value and indirectly to a stiffness value related to a specific drift at yield of the first storey of the retrofitted building.
Aiming at a target drift at yield of the first storey equal to ID y,target,1 =0.50%, with a ductility demand limited to µ target =2 (1.00% curve in the IDS of Fig. 13(b) for Ω 1 w =0), the required stiffness coefficient is equal to Ω 1 =λ c 2 Ω 1 J =8.52x10 -5 (Eq. 20(a)). The stiffness demand of the first storey is calculated according to Eq. (16) and is equal to corresponds to target period T target (=2π√(mB 2 /K 1 )=0.67 sec (where m=60.1 kN/(m/sec 2 ), Fig. 13(a)). Assuming that the cracked stiffness is equal to 50% of the elastic stiffness (EC8-I 2004), then the average λ factors for columns and beams are λ c =0.7 and λ b =0.3, respectively, whereas stiffness coefficient for beams is estimated equal to The vertical members of the first storey are designed for target stiffness K 1 =52710 kN/m, whereas the target stiffness of the other storeys is estimated from Fig. 3(a) by taking into account the stiffness ratios, κ i (K 2 =47439 kN/m, K 3 =36897 kN/m, K 4 =21084 kN/m). The target stiffness of each storey is equally distributed to the five columns of the storey (Table 2).
A note of caution is that detailing of the retrofitted cross-sections ought to comply with code provisions (regarding minimum bar diameter, minimum longitudinal reinforcement ratio and minimum jacket thickness) and therefore deviations from the target shape may be imperative due to code or construction limitations. Here, the required reinforcement for jackets of 1% in the 4 thstorey columns is subject to the code's restriction for minimum distance between longitudinal reinforcement and bar diameter. This led to an increased stiffness of the typical fourth storey column (instead of 4217 kN/m to 6051 kN/m).
(2) RS2: RC infill walls and selective RC column jacketing The second retrofit scenario aims at a fundamental change of the response of the frame by creating a dual lateral load resisting system (addition of two infill walls in the external bays incorporating the existing columns, RC jacketing of the interior row of columns ( Fig. 12(b)).
Behavior of the retrofitted structure is influenced greatly by the presence of the infill RC walls and for this reason it is assumed at the onset of the retrofit design that the target response profile would be closer to the flexural type. The EC8-I Type I (2004) earthquake design hazard was used again to define demand ( Fig. 13(a)).
The target stiffness of the other storeys is estimated from Fig. 3(c) by taking into account the stiffness ratios, κ i (K 2 =128069 kN/m, K 3 =72452 kN/m, K 4 =37978 kN/m).

Post-retrofit assessment
Models of the retrofitted frames were constructed in the environment of ZeusNL, a nonlinear finite element analysis platform (Elnashai et al. 2002).
Pushover analysis: Inelastic static pushover analyses were performed for a target drift at ultimate equal to 3.5% of the building height (target lateral displacement at the top: 420 mm), with the exception of the existing frame case, where analysis was terminated at a drift of 1.4% (=168 mm top lateral displacement, at 20% strength loss). The response of the retrofitted frame solutions is plotted against the response of the original frame in Fig. 14(a).
A fundamental modification of the response is effected by both retrofit solutions considered in terms of stiffness, strength and deformation capacity.
Distribution of plastic-hinge formation at member ends are illustrated in Figs. 14(c) and (d) for two drift levels (0.5%, and 1.0%) for the RS1 and RS2 retrofit solutions, respectively. It is observed that both retrofit options adopted have changed the sequence of plastic hinge formation as compared with the existing frame ( Fig. 14(b)), leading to a more desirable pattern with plastic hinges distributed along the beam ends in the retrofitted structures and eliminating mechanism formations (i.e., simultaneous plastic hinges in both columns ends in a single floor).
Dynamic time-history analysis: Behavior of the retrofitted frames was assessed by performing inelastic time history analyses for a group of artificial records used in the fullscale test of the ICONS frame (Pinto et al. 2002) which are representative of a moderatelyhigh European seismic hazard scenario; the El-Centro ground motion (ELC180 1940) was also used. The duration of significant excitation in the artificial records was around 15 sec and the peak ground acceleration was 0.22g, 0.29g and 0.38g, for the 475, 975 and 2000 year return events, respectively. The total duration of the El Centro record (ELC180 1940) was 40 sec and the peak ground acceleration was 0.31g. In the following, the return period is used as an identification code for the artificial record considered.
using secant to yield measures for member flexural stiffness properties). The post-design fundamental shape actually serves as a lower bound to the actual response profile. With advancing nonlinearity, the effective λ goes further down tending to 0 (since the plastic hinges are located in beams, λ b =0) thereby rendering the structure more "flexural", as is evident also in the latter deflection profiles in Fig. 15.
In case of the RS2 option the displacement response profiles (gray lines) ranged between the triangular and the flexural target response shapes (Fig. 16) and in most cases were bundled between an average (of the flexural and the triangular) and the triangular target shape. This indicates that the response of the resulting dual system was moderated as compared to a purely wall-type solution.
Additional information regarding the comparison of lateral displacement and interstorey drift profiles between the existing frame and the retrofitted ones are depicted in  Table 3. The presence of walls controlled the lateral displacement of the building significantly (Table 3).

Yield Point Spectra:
The retrofit solutions are assessed by using the Yield Point Spectra (Aschheim and Black 2000) for the El Centro ground motion and the EC8-I design spectra. In case of the El Centro ground motion, ductility demand for retrofit solutions RS1 and RS2 is equal to 1.25 and 1.07, respectively ( Fig. 19(a)). The ductility demand imposed by the EC8-I (2004) design spectrum (the retrofit solutions were designed according to it) is 2.0 for RS1 and 1.2 for RS2 ( Fig. 19(b)) thereby verifying realization of the retrofit strategy's objectives.

SUMMARY AND CONCLUSIONS
A displacement-based design methodology for development of the rehabilitation strategy of existing RC buildings was presented. Central concept of the proposed retrofit strategy was modifying the stiffness distribution so as to match a predefined target fundamental response shape. This target profile effectively serves as a first order approximation; the final mode shape may deviate somewhat from the targeted one. Interstorey Drift Spectra (IDS) were derived and practical design charts were established to readily relate the technological characteristics of well-known global intervention methods to drift demands. Stiffness required in order to limit drift within acceptable levels was obtained from the IDS. The methodology was tested in an example case study of a four-storey four-bay frame where two alternative retrofit strategies were applied. Post-retrofit assessment including pushover and time-history analysis of the alternative retrofit solutions revealed the efficiency of the IDS representation as a practical design tool which facilitates direct insight into the interrelation between drift demand and the required dimensions and details of the retrofit scenario.

ACKNOWLEDGMENTS
In conducting this research the authors benefited from the support and access to the facilities of their respective institutions. The research conducted by the third author was sponsored by the Mid-America Earthquake Center, a US National Science Foundation Engineering Research Center, funded under grant EEC 97-01785.

NOTATION
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