overstrenght and force reduction factors for multistorey reinforced-concrete strurctures

OVERSTRENGTH AND FORCE REDUCTION FACTORS OFMULTISTOREY REINFORCED-CONCRETE BUILDINGS

A. S. ELNASHAI1* AND A. M. MWAFY2

1Department of Civil and Environmental Engineering, University of Illinois at Urbana–Champaign, Urbana, IL, USA2Civil Engineering Department, University of Zagazig, Egypt

SUMMARY

This paper addresses the issue of horizontal overstrength in modern code-designed reinforced-concrete (RC)buildings. The relationship between the lateral capacity, the design force reduction factor, the ductility level andthe overstrength factor are investigated. The lateral capacity and the overstrength factor are estimated by means ofinelastic static pushover as well as time-history collapse analysis for 12 buildings of various characteristicsrepresenting a wide range of contemporary RC buildings. The importance of employing the elongated periods ofstructures to obtain the design forces is emphasized. Predicting this period from free vibration analysis byemploying ‘effective’ flexural stiffnesses is investigated. A direct relationship between the force reduction factorused in design and the lateral capacity of structures is confirmed in this study. Moreover, conservativeoverstrength of medium and low period RC buildings designed according to Eurocode 8 is proposed. Finally, theimplication of the force reduction factor on the commonly utilized overstrength definition is highlighted.Advantages of using an additional measure of response alongside the overstrength factor are emphasized. This isthe ratio between the overstrength factor and the force reduction factor and is termed the inherent overstrength(�i). The suggested measure provides more meaningful results of reserve strength and structural response thanoverstrength and force reduction factors. Copyright 2002 John Wiley & Sons, Ltd.

1. INTRODUCTION

Notwithstanding the recent development of deformation-based design methods, conventional seismicdesign procedures in all modern seismic codes still adopt force-based design criteria. The basicconcept of the former is to design the structure for a target displacement rather than a strength level.Hence, the deformation, which is the major cause of damage and collapse of structures subjected toearthquakes, can be controlled during the design. Nevertheless, the traditional concept of reducing theanticipated seismic forces using a single reduction factor, to arrive at the design force level, is stillwidely utilized. This is because of the satisfactory performance of buildings designed to modern codesin full-scale tests and during recent earthquakes especially with regard to life safety. Seismic codesrely on reserve strength and ductility, which improves the capability of the structure to absorb anddissipate energy, to justify this reduction. Hence, the role of the force reduction factor and theparameters influencing its evaluation and control are essential elements of seismic design according tocodes.

The values assigned to the response modification factor (R) of the US codes (FEMA, 1997; UBC,1997) are intended to account for both reserve strength and ductility (ATC, 1995). The same allowance

THE STRUCTURAL DESIGN OF TALL BUILDINGSStruct. Design Tall Build. 11, 329–351 (2002)Published online in Wiley InterScience (www.interscience.wiley.com). DOI:10.1002/tal.204

Copyright 2002 John Wiley & Sons, Ltd. Received October 2000Accepted March 2001

* Correspondence to: A. S. Elnashai, Department of Civil and Environmental Engineering, University of Illinois at Urbana–Champaign, Urbana, IL 61801, USA. E-mail: [email protected]

for overstrength is quite obvious in the force reduction factor definition of the Canadian (CCBFC,1995), the New Zealand (SNZ, 1992) and the Japanese (IAEE, 1992) codes. For instance, a calibrationfactor (U), which accounts for an overstrength of 1�67, has been already introduced in Canada.Eurocode 8 (EC8; see CEN, 1994) definition of the force reduction factor (behaviour factor q) does notexplicitly account for reserve strength. This is also clear from the lower factors of of EC8 comparedwith the US codes. However, structural systems with lower levels of redundancy are assigned lowerforce reduction factors in EC8, hence it implicitly takes into consideration that some factors contributeto overstrength. It is worthy to note that redundancy is considered here as a parameter contributing tooverstrength, contrary to the proposal of ATC-19 (ATC, 1995), splitting R into three factors: strength,ductility and redundancy. If the force reduction factors of EC8 are dependent on overstrength, then thelatter should be estimated when evaluating the former. If not, then the force reduction factors proposedby EC8 should be regarded as equivalent global ductility factors (R�) and overstrength should beaccounted for additionally (Fischinger and Fajfar, 1990).

It is therefore accepted to include the effect of reserve strength in calibrating the force reductionfactor. However, a generally applicable and precise estimation of overstrength is difficult to determinesince many factors contributing to it involve uncertainties. The actual strength of materials,confinement effects, the contribution of nonstructural elements and the actual participation of somestructural elements such as reinforced-concrete slabs are factors leading to high uncertainties (Humarand Ragozar, 1996). However, not all factors contributing to overstrength are favourable. Flexuraloverstrength in the beams of moment-resisting frames may cause storey collapse mechanisms or brittleshear failure in beams. Nonstructural elements also may cause shear failure in columns or soft storeyfailure (Park, 1996). Moreover, the overstrength factor varies widely according to the period of thestructure, the design intensity level, the structural system and the ductility level assumed in the design.This compounds the difficulties associated with evaluating this factor accurately.

Given that the reduction in seismic forces via the R factor is justified by the ductile response and theunquantified overstrength of structures, the accurate evaluation and investigation of interrelationshipsbetween these quantities, which are still based on engineering judgement, is an essential and pressingobjective. The aim of the current study is to evaluate and clarify the above, using a set of 12 RCbuildings varying in characteristics. The buildings are designed according to EC8, representingmodern seismic codes. The degree of variation between these buildings is considered to be sufficient tocover a reasonable range of conventional medium-rise buildings. The work is part of an extensivestudy to calibrate the force reduction factors of conventional RC buildings. For the sake of brevity,only the results of the strength-dependent component of the force reduction factor are presented in thispaper. Comprehensive results of this study are given elsewhere (Mwafy, 2000).

2. DESCRIPTION AND MODELLING OF THE BUILDINGS

2.1. Structural Systems

Twelve structures are assessed in this study. The buildings are designed and detailed in accordancewith EC8 (CEN, 1994) as a typical modern seismic design code applicable to more than one countrywith various levels of seismicity, soil conditions and types of construction. All buildings are assumedto be found on competent soil type B (medium dense sand or stiff clays). The buildings can becategorized into the three basic structural configurations illustrated in Figure 1. The structuralcharacteristics of the assessment sample are varied to represent the most common types of RCbuildings. Different building heights (24–36 m), structural systems (moment-resisting frames andframe-wall systems) and degree of elevation irregularity are taken into consideration. For each of thethree basic configurations, four buildings are produced from combinations of two design ground

330 A. S. ELNASHAI AND A. M. MWAFY

Copyright 2002 John Wiley & Sons, Ltd. Struct. Design Tall Build. 11, 329–351 (2002)

accelerations (0�15 g and 0�30 g) with three design ductility levels (high, medium and low), as shownin Table 1. The selected combinations enable examining the response of buildings designed to thesame ground acceleration but for different ductility levels, and vice versa. The level of ductility variesfrom high, dictating rigorous standards on member detailing, to low, requiring no special detailing orcapacity design requirements. Generally, lower design forces are adopted as a result of increasing theability of the structure to exhibit more ductile behaviour and therefore dissipate more energy. Theforce reduction factors used in the design as well as the observed elastic fundamental periods, Telastic,obtained from elastic free vibration analyses are also shown in Table 1.

The same overall plan dimensions of 15 m � 20 m are utilized for the 12 buildings. The total heightsfor the three groups are 25�5 m, 36 m and 24 m, respectively. The bottom storey of the first group ofbuildings has a height of 4�5 m. All other storeys for this group and for groups 2 and 3 have equalheights of 3 m. Further vertical irregularity is introduced in the first group by the cut-off at the groundstorey of four perimeter columns. These columns are supported by long span beams, as shown inFigure 1(a). The lateral force resisting system for groups 1 and 2 is moment resisting frames, whereasgroup 3 is provided with a central core and perimeter moment-resisting frames. The core consists oftwo channel shear walls coupled in the Z direction at each storey level by a pair of beams. The floor

Figure 1. Plane and sectional elevation of the buildings: (a) group 1, 8-storey irregular frame (IF) buildings; (b)group 2, 12-storey regular frame (RF) buildings; (c) group 3, 8-storey regular frame-wall (FW) buildings; for more

details, see Table 1

OVERSTRENGTH AND FORCE REDUCTION FACTORS 331


system of the groups 1 and 2 is solid slabs, whereas waffle slabs are utilized in group 3. A characteristiccylinder strength of 25 N mm�2 and a yield strength of 500 N mm�2 are considered for concrete andsteel, respectively. Further information regarding the cross-sectional dimensions and reinforcementdetails of members can be found elsewhere (Fardis, 1994).

2.2 Modelling for Inelastic Analysis

The finite element structural analysis program ADAPTIC (Izzuddin and Elnashai, 1989) is utilized toperform the inelastic analyses. ADAPTIC has been developed at Imperial College for the nonlinearanalysis of two-dimensional and three-dimensional steel, reinforced concrete and composite structuresunder static and dynamic loading, taking into account the effects of geometric nonlinearities andmaterial inelasticity. The program has the feature of representing the spread of inelasticity within themember cross-section and along the member length by utilizing the fibre approach. It is capable ofpredicting the large inelastic deformation of individual members and structures. Eigenvalue, static anddynamic analysis facilities are available and have been thoroughly tested and validated over the past12 years on the member and structure levels (e.g. Elnashai and Elghazouli, 1993; Elnashai andIzzuddin, 1993; Broderick and Elnashai, 1994; Martinez-Rueda and Elnashai, 1997; Pinho, 2000). Adetailed description of available elements and material models in ADAPTIC is beyond the scope ofthis paper. Further information regarding the program and its validation can be found in theaforementioned references.

Detailed and efficient two-dimensional models have been utilized for the 12 buildings investigated.Two-dimensional representation is selected owing to the limited significance of torsional effects forthe cases considered. With reference to the global axes system, the analyses are conducted along theglobal X axis for group 1 and 2, and the Z axis for group 3. Performing the analysis in theaforementioned directions can be justified by the fact that critical response criteria were expected tooccur earlier in those directions. The domination of gravity loads in the long span beams of the framestructural systems and the large amount of energy expected to be dissipated in the coupling beams of

Table 1. Structural systems considered

Reference no. Ductility level Design PGA (g)Force reduction

factor Telastic (s)

Group 1: 8-storey, irregular frame (IF) buildings:IF-H030 High 0�30 4�00 0�674IF-M030 Medium 0�30 3�00 0�654IF-M015 Medium 0�15 3�00 0�719IF-L015 Low 0�15 2�00 0�723

Group 2: 12-storey, regular frame (RF) buildings:RF-H030 High 0�30 5�00 0�857RF-M030 Medium 0�30 3�75 0�893RF-M015 Medium 0�15 3�75 0�920RF-L015 Low 0�15 2�50 0�913

Group 3: 8-storey, regular frame-wall (FW) buildings:FW-H030 High 0�30 3�50 0�538FW-M030 Medium 0�30 2�625 0�533FW-M015 Medium 0�15 2�625 0�592FW-L015 Low 0�15 1�75 0�588

Note: PGA, peals ground acceleration; Telastic, elastic fundamental period.



the frame-wall group supported this decision. Combination of the internal and the external lateral forceresisting systems is achieved by means of an overlay approach where the internal and the externalframes are coupled by rigid joint elements representing the high in-plane stiffness of the floor system.Figure 2 depicts the technique utilized for modelling the buildings.

Cubic shape function elements capable of representing the distribution of inelasticity are used tomodel the horizontal and vertical structural members. For this type of element, ADAPTIC performsthe numerical integration over two Gauss sections. Each Gauss section is subdivided into a number offibres where stresses and strains are calculated by applying the inelastic cyclic constitutiverelationships for each of the considered materials. Figure 2 illustrates the locations of the Gausssections within each element and the decomposition of a typical RC beam cross-section into confinedand unconfined areas. The concrete is represented in the current study by using a uniaxial constantconfinement concrete model (Martinez-Rueda and Elnashai, 1997). The advanced multisurfaceplasticity model (Elnashai and Izzuddin, 1993) is utilized for modelling the reinforcement bars. In thismodel, the stress–strain response of steel with nonlinear hardening and cyclic degradation is defined interms of a series of cubic polynomial functions. Mean values of material strength are utilized in theanalyses rather than the values used in the design; an approach consistent with ‘assessment’.

Three elements are utilized to model each horizontal and vertical structural member. The lengths ofthese elements are determined in accordance with the distribution of transverse and longitudinalreinforcements. Two rigid elements are utilized to connect the beam ends with the framing columns, asshown in Figure 2(a). Two shear spring elements are introduced to represent the shear stiffness of thebeam–column connection. In frame-wall buildings, the core wall on each side of the coupling beam ismodelled as ‘wide-column’. The elements are located at the centroid of the core U-shaped cross-section and connected with beam ends at each storey level using rigid arms. Bidiagonal reinforcementsof coupling beams are taken into consideration by adding the horizontal and the vertical projection ofthe steel area to the longitudinal and transverse reinforcement areas.

Gravity loads are applied as point loads at beam nodes. To account for inertia effects duringdynamic analysis, masses are distributed in the same pattern adopted for the gravity loads and arerepresented by lumped two-dimensional mass elements. The numerically dissipative Hilber–Hughes–Taylor �-integration scheme (Broderick, Elnashai and Izzuddin, 1994) is utilized to solve the equationsof motion.

2.3. Response Parameters

To assess the seismic performance of the 12 buildings and to obtain accurate analytical predictions ofthe reserve strength and the force reduction factor from inelastic static and dynamic analysis results,rigorous definitions of response parameters is needed. Two particularly important limit states in theresponse of the buildings are required for the approach utilized in this study; that at which significantyield occurs and that at which the first indication of failure is observed.

Two criteria are selected to define yield on member and structure levels. For adequately designedRC buildings, local yield is assumed when the strain in the main longitudinal tensile reinforcementexceeds the yield strain of steel. On the structure level, an elastic–perfectly-plastic idealisation of thereal system is employed, since no clear yield point is present. The initial stiffness is evaluated as thesecant stiffness at 75% of the ultimate strength, and the ultimate lateral strength of the real system isutilized for the post-elastic domain of the linearized envelope (Park, 1988). The top displacementcorresponding to the starting point of the post-elastic branch is obtained for each building from theinelastic pushover analysis and is employed in time-history analyses as the global yield limit state.

Two failure criteria on the member level are employed, exceeding the shear strength or the ultimatecurvature in any structural member. The shear supply of structural members is estimated mainly using



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the model proposed by Priestley, Verma and Xiao (1994). This model, which takes into account theinstantaneous influence of axial load and flexural ductility, provides an experimentally verifiableestimate of shear in RC members. To allow for effective comparison with the design code, the code-based shear strength model has also been employed after eliminating the safety factors. Furtherinformation regarding the implementation of the two shear models employed and the results of theextensive shear supply–demand assessment can be found elsewhere (Mwafy, 2000).

By considering overall structural characteristics, four criteria on the structure level are utilized todefine significant failure. The interstorey drift (ID) ratio is considered as the primary and mostimportant global collapse criterion. Several values for the ID collapse limit have been suggested in theliterature (SEAOC, 1995; FEMA 1996; Broderick and Elnashai, 1996). An upper limit of ID equal to3% is employed in this study. This limit should be sufficient to restrict second order (P–�) effects andto express the damage in structural and nonstructural elements. This limit is adopted over otherconservative limits to reflect the ability of structural frame systems to sustain relatively largedeformations, especially those designed to modern seismic codes such as the Uniform Building Code(UBC) or EC8. In additon, collapse also corresponds to the formation of a column hinging mechanismor a drop in the overall lateral resistance (by more than 10%). The interstorey drift sensitivitycoefficient (� = ID � storey gravity load/seismic storey shear) recommended by EC8 is utilized aswell. This criterion is intended to place a further check on second-order effects. Collapse is consideredto be imminent when this coefficient exceeds 0�3.

2.4. Selection and Normalization of Input Excitations for Dynamic Analysis

The lateral force profile utilized in pushover analysis was extensively investigated in another study(Mwafy and Elnashai, 2000). The design code load pattern is recommended for estimation of theseismic capacity of this set of buildings over the uniform load profile and the load obtained from modalanalyses (multimodal). The response of the 8-storey irregular frame buildings using the design codeload pattern, which is almost an inverted triangle, was identical to the response obtained from time-history collapse analysis. The investigation carried out on the 12-storey regular frame buildings andthe 8-storey hybrid structures also indicated that a conservative prediction of capacity and a reasonableestimation of deformation could be obtained using the same load distribution. Hence, this simple loadshape is employed in the current study.

Inelastic dynamic analysis for each building is performed using eight input excitations. Four 10-sduration artificially-generated records compatible with the EC8 elastic response spectrum for mediumsoil class (firm) were selected for comparison and calibration with the design code. Furthermore,previous analytical investigations and field evidence (Bozorgnia, Mahin and Brady, 1998; Papazoglouand Elnashai, 1996) have shown that the effect of the vertical component of the seismic excitation onstructural members and systems may be significant, particularly for buildings situated in the vicinity ofactive faults. For the set of structures considered, investigating this effect on the vertical vibrations ofthe planted columns of the irregular frame structures is notably important. It is also interesting toconsider the effect of utilizing the vertical component of ground motions on the shear supply–demandinvestigation carried out on structural members. Towards this end, two natural earthquakes wereselected in terms of the site-to-source distance and the vertical to horizontal(V/H) ratio and appliedwith and without the vertical ground motion component.

A total of 1500 inelastic time-history analyses were carried out on the detailed models described byusing the employed set of records to evaluate the overstrength and the force reduction factor. Owing tothe large number of analysis required if more natural records are employed (between 350 and 400analyses for each added natural record), the number of excitations was kept to this limit. The locationand characteristics of the selected records are given in Table 2. In Figure 3(a) the acceleration response



spectra for the artificial records and the longitudinal component of the natural records are comparedwith the EC8 elastic spectrum. The accelerograms and the code elastic spectrum are scaled to 0�30g,and the damping ratio set at 5%. The acceleration spectra for the vertical component of the two naturalevents are depicted in Figure 3(b) for the same peak ground acceleration (PGA).

The inelastic fundamental periods of vibration of the set of buildings were identified in anotherstudy (Mwafy and Elnashai, 2000) for eight seismic excitations and at different input intensities. Theaverages for the three groups of structure are 1�40, 1�75 and 0�9 seconds, respectively. In this periodrange all the spectral ordinates are comparable and the natural records envelope the code spectrum,contrary to the short period range, as shown in Figure 3(a). This is owing to the normalization methodadopted in the current study, where all records are scaled to possess equal velocity spectrum intensityin the period range utilized. In the short period range, it is clear that the spectral acceleration of theartificial and the Loma Prieta (SAR) records are significantly higher than Kobe (KBU), particularlyLoma Prieta (SAR), which will result in amplifying higher mode effects. Moreover, it is clear that the

Table 2. Ground motion records used in the time-history analysis

Kobe, Japan Loma Prieta, USA Artificial records

Station Kobe University Saratoga, ‘Aloha Ave.’ Art-rec1 to Art-rec4Date 17 January 1995 18 October 1989 Not applicableSurface Wave Magnitude Ms 7�20 7�17 Not applicableEpicentral distance (km) 14 17 Not applicable

Peals ground acceleration:horizontal, H 0�276 0�319 Not applicablevertical, V 0�431 0�349 Not applicable

V/H 1�56 1�09 Not applicableNo. of input excitations 2 2 4

Figure 3. Response spectra for 0�30g and 5% damping: (a) the artificial, the longitudinal component of the naturalaccelorograms and the EC8 (Eurocode 8, see CEN, 1994) spectrum; (b) the vertical component of the naturalrecords; group 1, 8-storey irregular frame (IF) buildings; group 2, 12-storey regular frame (RF) buildings; group 3,

8-storey regular frame-wall (FW) buildings; for more details, see Table 1



scaled spectral acceleration of the vertical component of the Lome Prieta (SAR) record is generallyhigher than the Kobe (KBU) spectrum with the exception of the period range 0�13–0�17 seconds.Therefore, it is expected that the effect of employing the vertical component of Lome Prieta (SAR) inthe analysis will be more noticeable than the Kobe (KBU) earthquake. Comprehensive results of theeffect of vertical ground motion on the seismic response of the buildings are presented and discussedelsewhere (Mwafy, 2000).

The technique adopted in this study is to subject each building to the selected set of excitations,which are scaled gradually upwards until all yield and collapse limit states are reached. Therefore, areliable scaling procedure is required. The employed scaling approach is based on the velocityspectrum intensity (Housner, 1952). The method is summarized as follows:

� Time-history collapse analyses are first carried out for the 12 buildings under the four artificialrecords, which are scaled according to their PGA. These records were generated to fit the codespectrum. Therefore, their velocity spectral intensity is equivalent to that of the code.

� The recorded top acceleration response is utilized to obtain the inelastic periods of each buildingusing a fast Fourier transform (FFT) algorithm taking the average for four artificial records.

� The normalization factor is the ratio SIc/SIn, where SIc and SIn are the areas under the code-impliedvelocity spectrum and the velocity spectrum of the scaled accelerogram, respectively. SIc and SIn arecalculated between periods of 0�8Ty and 1�2T2D, where Ty and T2D are defined as the inelastic periodsof the buildings at global yielding and at twice the design intensity, respectively. This limit has beenselected following extensive analysis and comparisons with different definitions of the scalingperiod range (Mwafy, 2000).

� The natural accelerograms are scaled gradually by utilizing the integration limits obtained for eachof the 12 buildings.

The average normalization coefficients at an intensity level of 0�30g alongside the inelastic periodrange employed to calculate the spectral areas for each group of building are shown in Table 3.

2.5. Analyses Performed

Eigenvalue analyses are conducted first to determine the uncracked horizontal and vertical periods ofvibration. It is also employed to estimate the inelastic period by reducing the flexural stiffness. Thissimple analysis is also useful as an initial validation tool of the analytical models. Inelastic staticpushover analyses are performed for the buildings by using an inverted triangular load. This analysisprocedure is employed to evaluate the global yield limit state, structural capacity and overstrength.Finally, extensive time-history collapse analysis is performed under the selected eight inputexcitations. This is carried out by progressively scaling and applying each accelerogram followed byassessment of the response and checking all performance criteria, starting from a relatively lowintensity, typically the design intensity divided by the force reduction factor, and ending with the

Table 3. Normalization factors for a peak ground acceleration of 0�30g

Natural record IF group (1�0–1�8 sec.)a RF group (1�3–2�4 sec.)a FW group (0�8–1�5 sec.)a

Kobe (KBU) 0�54 0�61 0�56Loma Prieta (SAR) 1�15 1�25 1�32

a :Integration limits (0�8Ty � 1�2T2D) used in the scaling technique.Note: IF, irregular frame; RF, regular frame; FW, regular frame-wall; for more details, see Table 1.



intensity at which all collapse definitions are achieved. Normally, between 15 to 20 analyses arerequired for each building–input-excitation combination to identify the response at different yield andcollapse limit states.

3. CONTRIBUTION OF THE ELONGATED PERIOD TO OVERSTRENGTH

3.1. Prediction of Inelastic Period from Eigenvalue Analysis

Eigenvalue analysis was utilized to investigate the possibility of defining an effective flexural stiffnessto predict the inelastic period of structures. Clearly, the reduction in the stiffness of the beams shouldbe higher than the columns as a consequence of applying capacity design rules. Moreover, the highcompressive axial forces in columns reduce the crack width and hence enhance the stiffness of thecracked section. Several values of the effective stiffness have been suggested in the literature. NEHRP(FEMA, 1997) recommends 0�5EcIg for beams and 0�7EcIg for columns, where EcIg is the stiffness ofthe uncracked concrete cross-section. Paulay and Priestley (1992) proposed 0�4EcIg for rectangularbeams and 0�35EcIg for T and L section beams. The effective stiffness of columns is suggested to rangefrom 0�4EcIg to 0�8EcIg according to the anticipated earthquake-induced axial force. The averagefundamental period for each group of building obtained from eigenvalue analyses using theaforementioned proposals are compared in Figure 4 with the identified elongation of the period duringtime-history analyses. The elongation of the period of the buildings has been evaluated (Mwafy andElnashai, 2000) using Fourier analyses of top acceleration response under eight seismic excitations, asexplained earlier.

For group 1, group 2 and group 3 of building the average elongation of the period at the design andtwice the design intensity levels are 88%–112%, 83%–100% and 45%–79%, respectively. Theobserved increase in the period obtained from eigenvalue analysis using the reduced stiffness valuesrecommended by FEMA 273 are 24%, 23% and 17%, respectively, whereas employing the suggestion

Figure 4. Comparison between the average inelastic period of the three groups at different intensity levels and theperiod obtained from eigenvalue analyses using different definitions of the effective stiffness; group 1, 8-storeyirregular frame (IF) buildings; group 2, 12-storey regular frame (RF) buildings; group 3, 8-storey regular frame-wall (FW) buildings; for more details, see Table 1; values of effective stiffness taken from FEMA, 1997; Paulay

and Priestley, 1992



of Paulay and Priestley shows an increase of 42%, 39% and 27%, respectively, for the three groups ofbuilding. It is clear that the recommendations of FEMA 273 and Paulay and Priestley underestimatethe inelastic period. It is also worth noting that the inelastic period at a low seismic intensity isgenerally close to the value at the design intensity. For instance, the average elastic period for the firstgroup of buildings is 0�69 sec, whereas that at half the design and at the design intensity levels are 1�20sec and 1�30 sec, respectively. This emphasises the overconservatism of the two proposals of FEMA273 and Paulay and Priestley (1992).

Utilizing half the effective stiffness values proposed by Paulay and Priestley (1992) increases theelastic periods by 75%, 71% and 52%, respectively, as shown in Figure 4. This seems to be morerealistic for frame structures. However, it is slightly unconservative for the frame-wall structures whencompared with the inelastic period at the design intensity level. For this type of structure, the responsedepends strongly on the walls. The cantilever response mode of these walls causes stress concentrationat the base and dissipation of a large amount of the energy imparted by the earthquake at this region.Therefore, the deterioration of the stiffness and the elongation of the period increase rapidly as a resultof extensive cracking and yielding at the base. The third group of buildings is reanalysed using half theeffective stiffness proposed by Paulay and Priestley for beams and columns, but the effective stiffnessof the walls are kept as suggested by these two researchers. The results illustrated in Figure 4 showconsistency with the results of the frame buildings in terms of the conservative and the rationalprediction of the inelastic period at the design intensity level.

3.2. Effect of Elongated Period on Overstrength

Whereas the above proposal for effective stiffness shows a conservative prediction of the inelasticperiod, it leads to significant reduction in design forces, as shown in Figure 5 for six of the investigatedbuildings. The design force values corresponding to the elastic period and the period obtained byutilizing the reduced stiffness are approximately represented by the size of the circles on the designspectra. It is observed that employing the elastic ‘uncracked’ period to obtain the design forces ratherthan the proposed definition of the effective stiffness increases the design forces by 43% and 19% for

Figure 5. Design spectra, elastic periods and cracked periods obtained using two definitions of the reducedstiffness for: (a) buildings designed to medium ductility and peak ground acceleration (PGA) = 0�30g; (b)buildings designed to low ductility and PGA = 0�15g; group 1, 8-storey irregular frame (IF) buildings; group 2, 12-storey regular frame (RF) buildings; group 3, 8-storey regular frame-wall (FW) buildings; for more details, see

Table 1



the frame and the frame-wall structures, respectively. It is also observed that employing the uncrackedperiod instead of the proposal of Paulay and Preistley (1992) leads to an increase in the design force by25% and 11% for the frame and the dual structures, respectively.

The above observations emphasize the importance of employing reduced stiffness when estimatingthe period of buildings where a noticeable saving in design forces and consequently in materials can beachieved. Conversely, overestimating the stiffness may lead to grossly unconservative estimates ofdeformations. This is avoided by employing realistic estimates of the inelastic period of vibration.Finally, it is important to note that, although the suggested reduction in the cross-section stiffness isreasonably conservative for the range of buildings investigated, more research is needed to coverbuildings out of this range, especially exceptionally short and long period structures. Moreover, thisproposal has been suggested to predict the inelastic period from eigenvalue analysis and hence isrestricted to this application. In another study, Li and Pourzanjani (1999) have shown that utilizing theproposal of Paulay and Priestley (1992) as well as the recommendation of FEMA 273 (FEMA, 1997)might be unconservative for predicting the displacement from time-history analyses.

4. RELATIONSHIP BETWEEN STRENGTH, THE FORCE REDUCTION FACTOR ANDOVERSTRENGTH

Previous research on the performance of buildings during severe earthquakes indicated that structuraloverstrength plays a very important role in protecting buildings from collapse. The overstrength factor(�d) may be defined as the ratio of the actual to the design lateral strength:

�d � Vy

Vd�1�

as depicted in Figure 6 and termed the ‘observed’ overstrength factor. Quantification of the actual

Figure 6. The relationships between the force reduction factor, R, structural overstrength, �, and the ductilityreduction factor, R�



overstrength can be employed to reduce the forces used in the design, hence leading to moreeconomical structures. The main sources of overstrength are reviewed in other studies (Uang, 1991;Mitchell and Paulter, 1994; Humar and Ragozar, 1996; Park, 1996). These include: (1) the differencebetween the actual and the design material strength; (2) conservatism of the design procedure andductility requirements; (3) load factors and multiple load cases; (4) accidental torsion consideration;(5) serviceability limit state provisions; (6) participation of nonstructural elements; (7) effect ofstructural elements not considered in predicting the lateral load capacity (e.g. actual slab width); (8)minimum reinforcement and member sizes that exceed the design requirements; (9) redundancy; (10)strain hardening; (11) actual confinement effect; and (12) utilizing the elastic period to obtain thedesign forces.

Figure 7. Comparison between the capacity envelopes for the three groups of buildings, obtained from inelasticstatic pushover analysis; Vy, actual strength (base shear); W, weight of building; �, top displacement; H, height ofbuilding; group 1, 8-storey irregular frame (IF) buildings; group 2, 12-storey regular frame (RF) buildings; group

3, 8-storey regular frame-wall (FW) buildings; for more details, see Table 1



4.1. Structural Capacity and Overstrength Analysis Results

Structural lateral capacity and overstrength can be well-assessed from inelastic analyses. These areestimated for the 12 buildings by means of inelastic static pushover as well as incremental time-historyanalyses up to collapse. The capacity envelopes of the buildings obtained from time-history collapseanalyses, which are presented elsewhere (Mwafy and Elnashai, 2000), are utilized to evaluateoverstrength factors. The envelopes were developed using regression analysis of the maximumresponse points (the roof displacement and the base shear) of eight seismic excitations for eachbuilding. Figure 7 shows the capacity envelopes for the three groups of building obtained frominelastic pushover analyses using a triangular lateral load distribution, where the base shear and the topdisplacement are normalized by the weight and the height of the building, respectively. The observedoverstrength factors from inelastic static pushover and time-history collapse analyses are depicted inFigure 8(a). The overstrength obtained from an alternative definition, as discussed hereafter, is shownin Figure 8(b). It is noteworthy that the maximum strength of the 12 buildings is generally observed ator before the interstorey drift collapse limit state, with the exception of few cases. In those cases thelateral strength at global collapse is considered as the maximum strength.

The point of first yield in horizontal and vertical structural members as well as the global yield limit,obtained from the elastic perfectly plastic idealized response, is illustrated on the response envelopesshown in Figure 7.

The interstorey drift collapse limit is also shown for each structure. For the second and third sets,first yield is observed in beams, whereas first yielding occurs at the second storey planted columns forthe irregular buildings. This is a result of the extra tensile forces imposed on these columns as a resultof the cut-off at the ground storey. This is clear in Figure 9, where the progress of hinge formation for asample building from each group is shown at the global yield limit state. The first observed 10 plastichinges can be distinguished in the figure from other subsequently formed hinges. The advantages ofemploying regular structural systems in the design are clear in Figures 7 and 9. The plastic hingesobserved in the second group of buildings up to the global yield limit state are mainly in beams. InFigure 9(b), only one plastic hinge in columns is observed (hinge 91 from a total of 96 hinges),

Figure 8. Observed (�d) and inherent (�i) overstrength for the 12 buildings; Vy, actual strength; Ve, elasticstrength; R, response modification factor; group 1, 8-storey irregular frame (IF) buildings; group 2, 12-storeyregular frame (RF) buildings; group 3, 8-storey regular frame-wall (FW) buildings; for more details, see Table 1



Figure 9. Progress in plastic hinge formation at the global yield limit state for a sample building from each group:(a) IF-M030 (top displacement = 289 mm, top drift ratio = 1�1%); (b) FR-M030 (top displacement = 359 mm, topdrift ratio = 1�0%); (c) FW-M030 (top displacement = 255 mm, top drift ratio = 1�1%); group 1, 8-storey irregularframe (IF) buildings; group 2, 12-storey regular frame (RF) buildings; group 3, 8-storey regular frame-wall (FW)

buildings; for more details, see Table 1



whereas several column and wall hinging cases are observed at this moderate level of lateral load in theirregular frame and the frame-wall structures. For the third group, first yield in vertical elements isobserved in the ground storey walls shortly after the first beam yielding. As explained earlier, thestiffness of these walls is significantly high compared with other lateral force resisting systems.Consequently, high levels of lateral force are attracted to the walls and high bending moments at theground level are generated as a result.

It is observed that more plastic hinges are formed in the external frames compared with the internalframes, particularly for the first two groups. For the frame structures, this is attributed to the higherstiffness of the short span external beams, which attract higher forces. The design of the internal framebeams are also dominated by gravity loads; therefore the effect of the lateral loads are less significant.For hybrid structures, plastic hinges generally form in the coupling beams earlier than in other internalbeams, reflecting the high energy dissipation potential of these members.

4.2. Lateral Capacity and Design Force Reduction Factor

Comparison between the capacity envelopes of the three sets (see Figure 7) shows that the highest V/Wratios are observed for the 8-storey frame-wall structures, reflecting the high stiffness and theefficiency of this structural system in resisting lateral forces. The lowest V/W values are observed forthe 12-storey regular frame buildings as a result of the high total gravity load. However, the maximumbase shear of the latter ranges from 7300 kN to 13200 kN, slightly higher than the maximum base shearobserved for the 8-storey irregular frame buildings (6600–12700 kN). The severance of four perimetercolumns and increasing the height of the ground storey leads to the observed reduction in the lateralstiffness of the irregular buildings.

Results for the pairs of buildings designed to the same seismic intensity and different levels ofductility show that the capacity is proportional to the code-defined force reduction factor. Table 4confirms that the increase in the capacity of lower ductility level buildings compared with their higherductility level counterparts is consistent with the reduction in the R factors adopted in the design. Theonly exception is the difference between the 0�30g design ground acceleration pair of the RF-group. Itis worth mentioning that in the three groups of buildings equal cross-sectional dimensions wereutilized with the buildings designed to the same seismic intensity with the exception of this pair, wherethe depth of beams of the RF-H030 building was increased to fulfil local ductility requirements of EC8(in terms of the allowable maximum tension reinforcement ratio within the critical regions of thebeams). It is concluded that a direct relationship between the lateral capacity and the design seismicforces can be applied to buildings that have the same section dimensions.

The difference in the lateral capacity between the two buildings designed to the same ductility level(ductility medium) and two different design intensities (0�30g and 0�15g) is also consistent with the

Table 4. Comparison of force reduction factor (R) and capacity (Vy) of buildings designed for equal seismicintensity and different ductility levels

0�30g pair 0�15g pair

reduction in R (%) increase in Vy (%) reduction in R (%) increase in Vy (%)

IF-group 25 25 33 26RF-group 25 8 33 34FW-group 25 26 33 26

Note: IF, irregular frame (group 1); RF, regular frame (group 2); FW, regular frame-wall (group 3).



difference in the design force levels. This is clear in Figure 7, where the lateral capacity of the M015buildings is almost half that of the M030 buildings (the highest and the lowest envelope). This issummarized in Table 5. It is important to point out that this observation cannot be generalized sincecross-sectional dimensions are not equal for buildings designed for different seismic intensity. This isclear from the response of the two buildings designed with ‘high’ and ‘low’ ductility in each group.The two buildings are designed for seismic intensity 0�30g and 0�15g, and the force reduction factor ofH030 is twice the value of that of L015. Hence, the design forces are equivalent for the two buildings.The difference in the lateral capacity shown in Figure 7 is the result of increasing the cross-sectionaldimensions of the H030 buildings. The aforementioned observations show clearly the trade-offbetween strength and ductility, through the force reduction factor.

In another study, Mitchell and Paulter (1994) have commented that it is a common misconceptionthat a structure designed with a force reduction factor R = 4�0 would have half the capacity of anotherdesigned with R = 2�0. It is noted in the aforementioned study that member cross-sectional sizes andlongitudinal reinforcements were higher in the buildings designed with a higher ductility level,particularly in columns and walls. This is reflected in the high overstrength factors calculated for thosebuildings (4�6 and 3�5) compared with lower ductility level buildings (2�14 and 2�78). This may beattributed to particular requirements of the Canadian design code (CCBFC, 1995). However, the directrelationship between the lateral capacity and the force reduction factor is explicitly confirmed in thepresent study when member cross-sectional sizes are kept constant.

4.3. Structure Overstrength (�d)

The estimated overstrength factors (�d) depicted in Figure 8(a) show that the values of overstrengthobtained from dynamic-to-collapse analyses are higher than those obtained from inelastic pushoveranalyses. This is a result of the conservatism of the design code lateral load distribution in predictingthe strength capacity of buildings (Mwafy and Elnashai, 2000). All the 12 buildings studied haveoverstrength factors over 2�0. The group-3 buildings exhibit the highest level of overstrength, theresults of the group-1 and group-2 buildings being comparable, particularly the factors obtained frompushover analyses. For the two buildings designed to the same ductility level in each group, the onedesigned to a lower seismic intensity exhibits higher overstrength, reflecting the higher contribution ofgravity loads. Higher ductility level buildings display higher reserve strength. This is attributed to theuse of a higher force reduction factor with these buildings, which causes a reduction in the designforces thus magnifying the effect of gravity loads. The rigorous provisions imposed on these buildingsto enhance ductility also lead to increased overstrength. Clearly, the observed reserve strength of the12 buildings is high, confirming the conservatism of EC8. The actual overstrength factors are expectedto be higher than the estimated values from static pushover or even time-history collapse analysisbecause of to the beneficial effects of some parameters that contribute to overstrength such asnonstructural elements.

Table 5. Comparison between the design forces and the lateral capacity of buildings designed with an equal forcereduction factor and different seismic forces (medium ductility buildings)

Reduction in design force level (%) Reduction in capacity (%)

IF-group 50 48RF-group 50 44FW-group 50 52

Note: IF, irregular frame (group 1); RF, regular frame (group 2); FW, regular frame-wall (group 3).



It is also clear in Figure 7 for the 12 buildings that the strength at first indication of member yielding(Vfy) is notably higher than the design strength levels (Vd) (refer also to Figure 6). The average Vfy/Vd

ratio for the three groups is 1�33, 1�46 and 1�57, respectively. Clearly, this ratio is relatively high,particularly for regular buildings. The main advantage of designing the structure at the design strengthlevel is to avoid explicit nonlinear structural analyses in the design and allow utilization of elasticanalysis methods (CEN, 1994; Uang 1991). It is observed in Figure 7 that the response does not showany significant deviation from the elastic behaviour at first yielding. Hence, this level can be safelyreduced to the design level. The aforementioned observation reflects the high reserve strength of EC8-designed structures.

Studies carried out on buildings designed to US seismic codes have indicated that the overstrengthfactor varied widely depending on the height of the building, the design seismic intensity and thestructural system. A brief review of these studies can be found in Jain and Navin (1995) and inWhittaker, Hart and Rojahn (1999). The scatter of the overstrength results was high, ranging from 1�8to 6�5 for long and short period range structures, respectively. Some extreme values are also reportedin the literature for low-rise buildings sited in low seismic zones.

From the few studies carried out on buildings designed according to EC8, Fischinger, Fajfar andVidie (1994) have studied 12 RC buildings to derive overstrength spectra. This has been proposed as afunction of the period and the ductility level. The effect of the design intensity and the structuralsystem in the determination of the reserve strength has not been accounted for. Overstrength factorsvaried from 1�6 for a 10-storey low ductility level building to 4�6 for a 3-storey high ductility structure.It is noted that the overstrength factors suggested in Fischinger and co-workers’ study are lower thanthe factors determined from the present investigation by about 30%. This is mainly attributed to thepredefined drift limit (1% of the building height) at which the strength of the buildings is calculatedfrom pushover analysis in Fischinger and co-workers’ study. It is clear in Figure 7 that the maximumstrength for the currently investigated buildings is observed at higher drift than the limit employed byFischinger, Fajfar and Vidie (1994). No strength degradation was considered in the analytical modelsutilized in their study. Therefore, overstrength was calculated at this conservative drift limit because ofthe uncertainty involved in identifying the maximum strength.

Panagiotakos and Fardis (1998) investigated three groups of frame structural systems of variousheights. Dynamic analysis was carried out only at the design intensity scaled by 1�0, 1�5 and 2�0.Overstrength factors ranging from 2�0 to 2�5 were observed at twice the design PGA for buildings with‘medium’ and ‘high’ ductility levels. Although, the ultimate capacity may not be attained at this levelof ground motion, the results of their study are consistent with the results of the current investigation interms of the high overstrength factors of EC8-designed buildings.

Developing overstrength spectra for RC buildings covering different structural systems, designintensity level, heights and ductility levels requires investigating a considerable range of buildingsrealistically designed and practically detailed to the code considered. As long as the above is notfulfilled, conservative overstrength for different classes of structures may be set. The rationality andthe conservatism of the minimum overstrength factor of 2�0 observed for the investigated sample ofmedium-rise RC buildings is supported by the following points:

� The average contribution to overstrength from the difference between mean and characteristicvalues of material strength exceeds 1�5. Employing the elastic period in the design adds tooverstrength factors by 1�19–1�43, according to the results shown in Figure 5. When designing underunidirectional seismic excitation, EC8 imposes an extra overstrength factor of 1�3 on columns sincethey are designed under biaxial bending. Consequently, the total overstrength from the aboveparameters is in excess of 2�0.

� The studies carried out on buildings designed to the US codes indicated a minimum overstrength of



1�8 for long period buildings. Reserve strength of conventional buildings is higher than this level.Overstrength of EC8-designed buildings should be higher than those of buildings designed to UScodes since the force reduction factors of EC8 are lower than those of the US codes.

� Actual overstrength should be higher than the values obtained from inelastic pushover analyses as aresult of conservatism of the code lateral load distribution in estimating the capacity andcontribution of nonstructural elements, as explained earlier.

It is also important to note that previous studies have confirmed that low-rise buildings exhibit higheroverstrength compared with medium-rise buildings. Therefore the minimum overstrength of 2�0 canalso be applied to this class of building. Moreover, seismic forces generally play a less important rolein the determination of cross-sectional sizes and reinforcements than do gravity loads, which governthe design of those buildings. Hence a precise evaluation of overstrength for the purpose of assessingforce reduction factors of short period buildings is not of great benefit. Finally, concerning long periodRC buildings, it is noteworthy that the overstrength of those buildings might be higher than the valuesobserved for medium-rise buildings since the design is likely to be governed by stiffness (or storeydrift; Uang, 1991).

4.4. Implication of the Force Reduction Factor on Overstrength Definition

Although the aforementioned definition of overstrength is widely utilized, it fails to indicate particularimportant features of seismic response. For instance, it is expected to observe higher overstrength forthe 8-storey irregular frame buildings compared with the 12-storey structures. The design code is morestringent for irregular structures and gravity loads play a more significant role in the design of lowerrise buildings. This causes relatively higher section dimensions and reinforcement ratios than thedemand imposed by seismic forces; hence higher overstrength is anticipated.

For two buildings designed to the same seismic intensity (elastic force, Ve) and different ductilitylevels, each structure will be assigned a different design force level (Vd) as a result of employing adifferent force reduction factor (R). Thus,

Vd1 � Ve

R1�2�

and

Vd2 � Ve

R2�3�

Therefore, the R factor assumed in the design is included in the overstrength (�d) calculated for thetwo buildings. Where,

�d1 � Vy1

Vd1�4�

and

�d2 � Vy2

Vd2�5�

The generic definition of overstrength (�d) is the ratio between actual (Vy) and design (Vd) strength. An



additional measure relating the actual (Vy) to the elastic strength level (Ve) is suggested for usealongside the overstrength (�d). The proposed measure (�i) may be expressed as:

�i � Vy

Ve� �d

R�6�

This definition, which is the inverse of the ductility part of the force reduction factor (R�) as shown inFigure 6, utilizes Ve. Hence the force reduction factor, which is defined on as empirical basis, isavoided in this definition. The suggested measure of response (�i) reflects the reserve strength and theanticipated behaviour of the structure under the design earthquake, as depicted in Figure 10. It istermed as ‘inherent’ to distinguish it from the ‘observed’ overstrength commonly used in the literature.Clearly, in the case of �i � 1�0 the global response will be almost elastic under the design earthquake,reflecting the high overstrength of the structure. Below this limit, the difference between the value of�i and unity is an indication of the ratio of the forces that are imposed on the structure in the postelasticrange.

The results of the proposed measure (�i) are shown in Figure 8(b) alongside the overstrength factors(�d) to facilitate comparison between the two measures. It is clear that the values of �i are quite highfor the third group of buildings. The strength levels of the four buildings of this group exceed theelastic strength, with the exception of the results of the pushover analysis for the FW-H030 building.The proposed measure clearly reflects the overconservatism of EC8 for structural wall systems, whereminimum section sizes and reinforcements lead to an elastic response for this class of structure underthe design intensity. Moreover, it is clear that the response of the buildings designed to a low ductilitylevel in each group are likely to be elastic, which again reflects the conservatism of the code. It is worthmentioning that for such a type of structure no capacity design rules are applied, although somerequirements to enhance the ductility are imposed. The lowest force reduction factors of EC8 areassigned to this class of building (1�5–2�5). Even though the buildings designed to this ductility levelare not intended to respond well into the postelastic range or dissipate significant amounts of energy,the actual strength of these buildings is relatively high. This implies that the level of ductility has notbeen exploited by the design code to reduce the design forces through employing higher forcereduction factors.

Contrary to the conventional definition of overstrength (�d), the results for �i display clearly theexpected higher overstrength of the IF group of buildings compared with the RF group, as shown in

Figure 10. Different levels of the inherent overstrength, �i: (a) ductile response, �i � 1�0; (b) elastic responseunder the design earthquake, �i � 1�0; Vd, design strength; Vy, actual strength; Ve, elastic strength; Disp.,

displacement



Figure 8. However, for the buildings designed to the same seismic intensity in each group, the higherductility level buildings show lower values of �i, reflecting the higher ratio of the forces that areimposed on the structure in the postelastic range. Finally, the values of �i are consistent with theresults of the overstrength (�d) in terms of the observed higher values for the buildings designed tolower seismic intensity.

5. CONCLUSIONS

In this paper, 12 RC buildings with various characteristics were studied to evaluate the overstrengthand investigate its relationship with the force reduction factor. The study was carried out by means ofeigenvalue, inelastic pushover and time-history collapse analysis employing eight natural and artificialrecords. Response criteria at both member and storey levels were defined. The input accelerogramswere scaled to equal velocity spectrum intensity. Notwithstanding the limitations of using a finite set ofinput motions and specific structures, the following conclusions are applicable to a large class of RCbuilding.

� A conservative prediction of the elongated period of structures can be obtained by employing areduced flexural stiffness in eigenvalue analysis. Using the effective stiffness values suggested byFEMA 273 (FEMA, 1997) and Paulay and Priestley (1992) in eigenvalue analysis underestimatesthe inelastic periods. Employing the values suggested by Paulay and Priestley for walls and half ofthose values for beams and columns show a reasonably conservative prediction of the cracked periodat the design intensity. Utilizing this period in the design instead of the elastic period leads to areduction in the design forces of 15%–30%.

� A direct relationship between the lateral capacity and the design seismic force is confirmed here.This can be applied to buildings that share section dimensions.

� High overstrength factors are exhibited for the sample studied, which has elastic and inelasticperiods of 0�53–0�92 and 0�9–1�75, respectively. The minimum observed overstrength factor is 2�0.The buildings designed to low seismic intensity levels show high overstrength factors as a result ofthe dominant role of gravity loads. Hybrid structural systems and buildings designed to high ductilitylevels also exhibit high overstrength. This is a result of minimum cross-section—sizes andreinforcements, which are more stringently applied to such buildings.

� The contribution to overstrength from three sources exceeds a factor of 2�0 These are: (1) differencebetween mean and characteristic values of material strength, (2) employing the elastic period in thedesign instead of the cracked period and (3) designing the columns in biaxial bending when analysedunder unidirectional seismic excitation.

� Overstrength during earthquakes should be higher than the values obtained from inelastic staticanalyses using the code lateral load distribution. The triangular load is conservative in predicting theultimate capacity. Also, contributions of nonstructural elements should produce higher capacity andhence higher overstrength.

� If overstrength is not accurately evaluated by means of inelastic analysis, a lower bound may beutilized. A conservative overstrength factor of 2�0 is suggested for medium period RC buildingsdesigned and detailed to EC8 (in principle, this applies to other modern codes). This limit can beapplied to low-rise buildings since they usually possess higher overstrength than do medium-risebuildings.

� It is suggested to utilize an additional measure of response alongside the widely utilized overstrengthmeasure (�d). This is herein termed the ‘inherent overstrength factor’ �i. The former factor relatesthe ultimate strength to the force assumed in the design, whereas the proposed measure relates theultimate strength to the elastic force. This avoids implicating the force reduction factor in thedefinition. The overstrength measure is needed for evaluation and possible calibration of the force



reduction factor adopted in the design. However, it may lead to unreliable predictions ofoverstrength because of the inclusion of the force reduction factor assumed in the design in itsdefinition. It also failed to confirm clearly the conservatism of the code since the range of itsvariation is too wide. The suggested measure (�i) better reflects the anticipated behaviour of thestructure and the reserve strength under the design earthquake owing to its explicit elastic responselimit.

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