The Structural Design Of Tall Buildings

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

OVERSTRENGTH AND FORCE REDUCTION FACTORS OF MULTISTOREY REINFORCED-CONCRETE BUILDINGS A. S. ELNASHAI1* AND A. M. MWAFY2 1

Department of Civil and Environmental Engineering, University of Illinois at Urbana–Champaign, Urbana, IL, USA 2 Civil 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 and the overstrength factor are investigated. The lateral capacity and the overstrength factor are estimated by means of inelastic static pushover as well as time-history collapse analysis for 12 buildings of various characteristics representing a wide range of contemporary RC buildings. The importance of employing the elongated periods of structures to obtain the design forces is emphasized. Predicting this period from free vibration analysis by employing ‘effective’ flexural stiffnesses is investigated. A direct relationship between the force reduction factor used in design and the lateral capacity of structures is confirmed in this study. Moreover, conservative overstrength of medium and low period RC buildings designed according to Eurocode 8 is proposed. Finally, the implication 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 is the 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 than overstrength and force reduction factors. Copyright  2002 John Wiley & Sons, Ltd.

1.

INTRODUCTION

Notwithstanding the recent development of deformation-based design methods, conventional seismic design procedures in all modern seismic codes still adopt force-based design criteria. The basic concept 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 to earthquakes, can be controlled during the design. Nevertheless, the traditional concept of reducing the anticipated seismic forces using a single reduction factor, to arrive at the design force level, is still widely utilized. This is because of the satisfactory performance of buildings designed to modern codes in full-scale tests and during recent earthquakes especially with regard to life safety. Seismic codes rely on reserve strength and ductility, which improves the capability of the structure to absorb and dissipate energy, to justify this reduction. Hence, the role of the force reduction factor and the parameters influencing its evaluation and control are essential elements of seismic design according to codes. 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

* 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]

Copyright  2002 John Wiley & Sons, Ltd.

Received October 2000 Accepted March 2001

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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 calibration factor (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 not explicitly account for reserve strength. This is also clear from the lower factors of of EC8 compared with the US codes. However, structural systems with lower levels of redundancy are assigned lower force reduction factors in EC8, hence it implicitly takes into consideration that some factors contribute to overstrength. It is worthy to note that redundancy is considered here as a parameter contributing to overstrength, 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 the latter should be estimated when evaluating the former. If not, then the force reduction factors proposed by EC8 should be regarded as equivalent global ductility factors (R ) and overstrength should be accounted for additionally (Fischinger and Fajfar, 1990). It is therefore accepted to include the effect of reserve strength in calibrating the force reduction factor. However, a generally applicable and precise estimation of overstrength is difficult to determine since many factors contributing to it involve uncertainties. The actual strength of materials, confinement effects, the contribution of nonstructural elements and the actual participation of some structural elements such as reinforced-concrete slabs are factors leading to high uncertainties (Humar and Ragozar, 1996). However, not all factors contributing to overstrength are favourable. Flexural overstrength in the beams of moment-resisting frames may cause storey collapse mechanisms or brittle shear failure in beams. Nonstructural elements also may cause shear failure in columns or soft storey failure (Park, 1996). Moreover, the overstrength factor varies widely according to the period of the structure, 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 the unquantified overstrength of structures, the accurate evaluation and investigation of interrelationships between these quantities, which are still based on engineering judgement, is an essential and pressing objective. The aim of the current study is to evaluate and clarify the above, using a set of 12 RC buildings varying in characteristics. The buildings are designed according to EC8, representing modern seismic codes. The degree of variation between these buildings is considered to be sufficient to cover a reasonable range of conventional medium-rise buildings. The work is part of an extensive study 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 this paper. Comprehensive results of this study are given elsewhere (Mwafy, 2000).

2. 2.1.

DESCRIPTION AND MODELLING OF THE BUILDINGS

Structural Systems

Twelve structures are assessed in this study. The buildings are designed and detailed in accordance with EC8 (CEN, 1994) as a typical modern seismic design code applicable to more than one country with various levels of seismicity, soil conditions and types of construction. All buildings are assumed to be found on competent soil type B (medium dense sand or stiff clays). The buildings can be categorized into the three basic structural configurations illustrated in Figure 1. The structural characteristics of the assessment sample are varied to represent the most common types of RC buildings. Different building heights (24–36 m), structural systems (moment-resisting frames and frame-wall systems) and degree of elevation irregularity are taken into consideration. For each of the three basic configurations, four buildings are produced from combinations of two design ground Copyright  2002 John Wiley & Sons, Ltd.

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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

accelerations (0
15 g and 0
30 g) with three design ductility levels (high, medium and low), as shown in Table 1. The selected combinations enable examining the response of buildings designed to the same ground acceleration but for different ductility levels, and vice versa. The level of ductility varies from high, dictating rigorous standards on member detailing, to low, requiring no special detailing or capacity design requirements. Generally, lower design forces are adopted as a result of increasing the ability of the structure to exhibit more ductile behaviour and therefore dissipate more energy. The force 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 heights for the three groups are 25
5 m, 36 m and 24 m, respectively. The bottom storey of the first group of buildings has a height of 4
5 m. All other storeys for this group and for groups 2 and 3 have equal heights of 3 m. Further vertical irregularity is introduced in the first group by the cut-off at the ground storey of four perimeter columns. These columns are supported by long span beams, as shown in Figure 1(a). The lateral force resisting system for groups 1 and 2 is moment resisting frames, whereas group 3 is provided with a central core and perimeter moment-resisting frames. The core consists of two channel shear walls coupled in the Z direction at each storey level by a pair of beams. The floor Copyright  2002 John Wiley & Sons, Ltd.

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Table 1. Structural systems considered Reference no.

Ductility level

Force reduction factor

Telastic (s)

0
30 0
30 0
15 0
15

4
00 3
00 3
00 2
00

0
674 0
654 0
719 0
723

0
30 0
30 0
15 0
15

5
00 3
75 3
75 2
50

0
857 0
893 0
920 0
913

0
30 0
30 0
15 0
15

3
50 2
625 2
625 1
75

0
538 0
533 0
592 0
588

Design PGA (g)

Group 1: 8-storey, irregular frame (IF) buildings: IF-H030 High IF-M030 Medium IF-M015 Medium IF-L015 Low Group 2: 12-storey, regular frame (RF) buildings: RF-H030 High RF-M030 Medium RF-M015 Medium RF-L015 Low Group 3: 8-storey, regular frame-wall (FW) buildings: FW-H030 High FW-M030 Medium FW-M015 Medium FW-L015 Low

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

system of the groups 1 and 2 is solid slabs, whereas waffle slabs are utilized in group 3. A characteristic cylinder strength of 25 N mm 2 and a yield strength of 500 N mm 2 are considered for concrete and steel, respectively. Further information regarding the cross-sectional dimensions and reinforcement details 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 to perform the inelastic analyses. ADAPTIC has been developed at Imperial College for the nonlinear analysis of two-dimensional and three-dimensional steel, reinforced concrete and composite structures under static and dynamic loading, taking into account the effects of geometric nonlinearities and material inelasticity. The program has the feature of representing the spread of inelasticity within the member cross-section and along the member length by utilizing the fibre approach. It is capable of predicting the large inelastic deformation of individual members and structures. Eigenvalue, static and dynamic analysis facilities are available and have been thoroughly tested and validated over the past 12 years on the member and structure levels (e.g. Elnashai and Elghazouli, 1993; Elnashai and Izzuddin, 1993; Broderick and Elnashai, 1994; Martinez-Rueda and Elnashai, 1997; Pinho, 2000). A detailed description of available elements and material models in ADAPTIC is beyond the scope of this paper. Further information regarding the program and its validation can be found in the aforementioned 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 for the cases considered. With reference to the global axes system, the analyses are conducted along the global X axis for group 1 and 2, and the Z axis for group 3. Performing the analysis in the aforementioned directions can be justified by the fact that critical response criteria were expected to occur earlier in those directions. The domination of gravity loads in the long span beams of the frame structural systems and the large amount of energy expected to be dissipated in the coupling beams of Copyright  2002 John Wiley & Sons, Ltd.

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the frame-wall group supported this decision. Combination of the internal and the external lateral force resisting systems is achieved by means of an overlay approach where the internal and the external frames 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 to model the horizontal and vertical structural members. For this type of element, ADAPTIC performs the numerical integration over two Gauss sections. Each Gauss section is subdivided into a number of fibres where stresses and strains are calculated by applying the inelastic cyclic constitutive relationships for each of the considered materials. Figure 2 illustrates the locations of the Gauss sections within each element and the decomposition of a typical RC beam cross-section into confined and unconfined areas. The concrete is represented in the current study by using a uniaxial constant confinement concrete model (Martinez-Rueda and Elnashai, 1997). The advanced multisurface plasticity model (Elnashai and Izzuddin, 1993) is utilized for modelling the reinforcement bars. In this model, the stress–strain response of steel with nonlinear hardening and cyclic degradation is defined in terms of a series of cubic polynomial functions. Mean values of material strength are utilized in the analyses 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 of these elements are determined in accordance with the distribution of transverse and longitudinal reinforcements. Two rigid elements are utilized to connect the beam ends with the framing columns, as shown in Figure 2(a). Two shear spring elements are introduced to represent the shear stiffness of the beam–column connection. In frame-wall buildings, the core wall on each side of the coupling beam is modelled as ‘wide-column’. The elements are located at the centroid of the core U-shaped crosssection and connected with beam ends at each storey level using rigid arms. Bidiagonal reinforcements of coupling beams are taken into consideration by adding the horizontal and the vertical projection of the steel area to the longitudinal and transverse reinforcement areas. Gravity loads are applied as point loads at beam nodes. To account for inertia effects during dynamic analysis, masses are distributed in the same pattern adopted for the gravity loads and are represented by lumped two-dimensional mass elements. The numerically dissipative Hilber–Hughes– Taylor a-integration scheme (Broderick, Elnashai and Izzuddin, 1994) is utilized to solve the equations of motion. 2.3.

Response Parameters

To assess the seismic performance of the 12 buildings and to obtain accurate analytical predictions of the 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 the response of the buildings are required for the approach utilized in this study; that at which significant yield 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 designed RC buildings, local yield is assumed when the strain in the main longitudinal tensile reinforcement exceeds the yield strain of steel. On the structure level, an elastic–perfectly-plastic idealisation of the real system is employed, since no clear yield point is present. The initial stiffness is evaluated as the secant stiffness at 75% of the ultimate strength, and the ultimate lateral strength of the real system is utilized for the post-elastic domain of the linearized envelope (Park, 1988). The top displacement corresponding to the starting point of the post-elastic branch is obtained for each building from the inelastic 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 ultimate curvature in any structural member. The shear supply of structural members is estimated mainly using Copyright  2002 John Wiley & Sons, Ltd.

Struct. Design Tall Build. 11, 329–351 (2002)

Figure 2. Adopted modelling approach: (a) beam–column connection; (b) cubic elastoplastic element; (c) decomposition of beam T-section into fibres

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the model proposed by Priestley, Verma and Xiao (1994). This model, which takes into account the instantaneous influence of axial load and flexural ductility, provides an experimentally verifiable estimate of shear in RC members. To allow for effective comparison with the design code, the codebased shear strength model has also been employed after eliminating the safety factors. Further information regarding the implementation of the two shear models employed and the results of the extensive shear supply–demand assessment can be found elsewhere (Mwafy, 2000). By considering overall structural characteristics, four criteria on the structure level are utilized to define significant failure. The interstorey drift (ID) ratio is considered as the primary and most important global collapse criterion. Several values for the ID collapse limit have been suggested in the literature (SEAOC, 1995; FEMA 1996; Broderick and Elnashai, 1996). An upper limit of ID equal to 3% is employed in this study. This limit should be sufficient to restrict second order (P–D) effects and to express the damage in structural and nonstructural elements. This limit is adopted over other conservative limits to reflect the ability of structural frame systems to sustain relatively large deformations, 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 mechanism or a drop in the overall lateral resistance (by more than 10%). The interstorey drift sensitivity coefficient ( = ID  storey gravity load/seismic storey shear) recommended by EC8 is utilized as well. This criterion is intended to place a further check on second-order effects. Collapse is considered to 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 the seismic capacity of this set of buildings over the uniform load profile and the load obtained from modal analyses (multimodal). The response of the 8-storey irregular frame buildings using the design code load pattern, which is almost an inverted triangle, was identical to the response obtained from timehistory collapse analysis. The investigation carried out on the 12-storey regular frame buildings and the 8-storey hybrid structures also indicated that a conservative prediction of capacity and a reasonable estimation of deformation could be obtained using the same load distribution. Hence, this simple load shape is employed in the current study. Inelastic dynamic analysis for each building is performed using eight input excitations. Four 10-s duration artificially-generated records compatible with the EC8 elastic response spectrum for medium soil class (firm) were selected for comparison and calibration with the design code. Furthermore, previous analytical investigations and field evidence (Bozorgnia, Mahin and Brady, 1998; Papazoglou and Elnashai, 1996) have shown that the effect of the vertical component of the seismic excitation on structural members and systems may be significant, particularly for buildings situated in the vicinity of active faults. For the set of structures considered, investigating this effect on the vertical vibrations of the planted columns of the irregular frame structures is notably important. It is also interesting to consider the effect of utilizing the vertical component of ground motions on the shear supply–demand investigation carried out on structural members. Towards this end, two natural earthquakes were selected in terms of the site-to-source distance and the vertical to horizontal(V/H) ratio and applied with and without the vertical ground motion component. A total of 1500 inelastic time-history analyses were carried out on the detailed models described by using the employed set of records to evaluate the overstrength and the force reduction factor. Owing to the large number of analysis required if more natural records are employed (between 350 and 400 analyses for each added natural record), the number of excitations was kept to this limit. The location and characteristics of the selected records are given in Table 2. In Figure 3(a) the acceleration response Copyright  2002 John Wiley & Sons, Ltd.

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

Station Date Surface Wave Magnitude Ms Epicentral distance (km) Peals ground acceleration: horizontal, H vertical, V V/H No. of input excitations

Kobe, Japan

Loma Prieta, USA

Kobe University 17 January 1995 7
20 14

Saratoga, ‘Aloha Ave.’ 18 October 1989 7
17 17

0
276 0
431 1
56 2

0
319 0
349 1
09 2

Artificial records Art-rec1 to Art-rec4 Not applicable Not applicable Not applicable Not applicable Not applicable Not applicable 4

spectra for the artificial records and the longitudinal component of the natural records are compared with 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 natural events 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 another study (Mwafy and Elnashai, 2000) for eight seismic excitations and at different input intensities. The averages for the three groups of structure are 1
40, 1
75 and 0
9 seconds, respectively. In this period range 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 method adopted in the current study, where all records are scaled to possess equal velocity spectrum intensity in the period range utilized. In the short period range, it is clear that the spectral acceleration of the artificial and the Loma Prieta (SAR) records are significantly higher than Kobe (KBU), particularly Loma Prieta (SAR), which will result in amplifying higher mode effects. Moreover, it is clear that the

Figure 3. Response spectra for 0
30g and 5% damping: (a) the artificial, the longitudinal component of the natural accelorograms and the EC8 (Eurocode 8, see CEN, 1994) spectrum; (b) the vertical component of the natural records; 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 Copyright  2002 John Wiley & Sons, Ltd.

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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) Loma Prieta (SAR)

0
54 1
15

0
61 1
25

0
56 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.

scaled spectral acceleration of the vertical component of the Lome Prieta (SAR) record is generally higher 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) in the analysis will be more noticeable than the Kobe (KBU) earthquake. Comprehensive results of the effect of vertical ground motion on the seismic response of the buildings are presented and discussed elsewhere (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, a reliable scaling procedure is required. The employed scaling approach is based on the velocity spectrum intensity (Housner, 1952). The method is summarized as follows: . Time-history collapse analyses are first carried out for the 12 buildings under the four artificial records, which are scaled according to their PGA. These records were generated to fit the code spectrum. 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 building using 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-implied velocity spectrum and the velocity spectrum of the scaled accelerogram, respectively. SIc and SIn are calculated between periods of 0
8Ty and 1
2T2D, where Ty and T2D are defined as the inelastic periods of the buildings at global yielding and at twice the design intensity, respectively. This limit has been selected following extensive analysis and comparisons with different definitions of the scaling period range (Mwafy, 2000). . The natural accelerograms are scaled gradually by utilizing the integration limits obtained for each of the 12 buildings. The average normalization coefficients at an intensity level of 0
30g alongside the inelastic period range 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 of vibration. It is also employed to estimate the inelastic period by reducing the flexural stiffness. This simple analysis is also useful as an initial validation tool of the analytical models. Inelastic static pushover analyses are performed for the buildings by using an inverted triangular load. This analysis procedure 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 input excitations. This is carried out by progressively scaling and applying each accelerogram followed by assessment of the response and checking all performance criteria, starting from a relatively low intensity, typically the design intensity divided by the force reduction factor, and ending with the Copyright  2002 John Wiley & Sons, Ltd.

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Figure 4. Comparison between the average inelastic period of the three groups at different intensity levels and the period obtained from eigenvalue analyses using different definitions of the effective stiffness; group 1, 8-storey irregular frame (IF) buildings; group 2, 12-storey regular frame (RF) buildings; group 3, 8-storey regular framewall (FW) buildings; for more details, see Table 1; values of effective stiffness taken from FEMA, 1997; Paulay and Priestley, 1992

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

3. 3.1.

CONTRIBUTION OF THE ELONGATED PERIOD TO OVERSTRENGTH

Prediction of Inelastic Period from Eigenvalue Analysis

Eigenvalue analysis was utilized to investigate the possibility of defining an effective flexural stiffness to predict the inelastic period of structures. Clearly, the reduction in the stiffness of the beams should be higher than the columns as a consequence of applying capacity design rules. Moreover, the high compressive axial forces in columns reduce the crack width and hence enhance the stiffness of the cracked 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 of the uncracked concrete cross-section. Paulay and Priestley (1992) proposed 0
4EcIg for rectangular beams and 0
35EcIg for T and L section beams. The effective stiffness of columns is suggested to range from 0
4EcIg to 0
8EcIg according to the anticipated earthquake-induced axial force. The average fundamental period for each group of building obtained from eigenvalue analyses using the aforementioned proposals are compared in Figure 4 with the identified elongation of the period during time-history analyses. The elongation of the period of the buildings has been evaluated (Mwafy and Elnashai, 2000) using Fourier analyses of top acceleration response under eight seismic excitations, as explained earlier. For group 1, group 2 and group 3 of building the average elongation of the period at the design and twice the design intensity levels are 88%–112%, 83%–100% and 45%–79%, respectively. The observed increase in the period obtained from eigenvalue analysis using the reduced stiffness values recommended by FEMA 273 are 24%, 23% and 17%, respectively, whereas employing the suggestion Copyright  2002 John Wiley & Sons, Ltd.

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Figure 5. Design spectra, elastic periods and cracked periods obtained using two definitions of the reduced stiffness 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, 12storey regular frame (RF) buildings; group 3, 8-storey regular frame-wall (FW) buildings; for more details, see Table 1

of Paulay and Priestley shows an increase of 42%, 39% and 27%, respectively, for the three groups of building. It is clear that the recommendations of FEMA 273 and Paulay and Priestley underestimate the inelastic period. It is also worth noting that the inelastic period at a low seismic intensity is generally close to the value at the design intensity. For instance, the average elastic period for the first group of buildings is 0
69 sec, whereas that at half the design and at the design intensity levels are 1
20 sec and 1
30 sec, respectively. This emphasises the overconservatism of the two proposals of FEMA 273 and Paulay and Priestley (1992). Utilizing half the effective stiffness values proposed by Paulay and Priestley (1992) increases the elastic periods by 75%, 71% and 52%, respectively, as shown in Figure 4. This seems to be more realistic for frame structures. However, it is slightly unconservative for the frame-wall structures when compared with the inelastic period at the design intensity level. For this type of structure, the response depends strongly on the walls. The cantilever response mode of these walls causes stress concentration at 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 result of extensive cracking and yielding at the base. The third group of buildings is reanalysed using half the effective stiffness proposed by Paulay and Priestley for beams and columns, but the effective stiffness of the walls are kept as suggested by these two researchers. The results illustrated in Figure 4 show consistency with the results of the frame buildings in terms of the conservative and the rational prediction 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 inelastic period, it leads to significant reduction in design forces, as shown in Figure 5 for six of the investigated buildings. The design force values corresponding to the elastic period and the period obtained by utilizing the reduced stiffness are approximately represented by the size of the circles on the design spectra. It is observed that employing the elastic ‘uncracked’ period to obtain the design forces rather than the proposed definition of the effective stiffness increases the design forces by 43% and 19% for Copyright  2002 John Wiley & Sons, Ltd.

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

the frame and the frame-wall structures, respectively. It is also observed that employing the uncracked period instead of the proposal of Paulay and Preistley (1992) leads to an increase in the design force by 25% and 11% for the frame and the dual structures, respectively. The above observations emphasize the importance of employing reduced stiffness when estimating the period of buildings where a noticeable saving in design forces and consequently in materials can be achieved. Conversely, overestimating the stiffness may lead to grossly unconservative estimates of deformations. 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 is reasonably conservative for the range of buildings investigated, more research is needed to cover buildings out of this range, especially exceptionally short and long period structures. Moreover, this proposal has been suggested to predict the inelastic period from eigenvalue analysis and hence is restricted to this application. In another study, Li and Pourzanjani (1999) have shown that utilizing the proposal 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 AND OVERSTRENGTH

Previous research on the performance of buildings during severe earthquakes indicated that structural overstrength 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 ˆ

Vy Vd

…1†

as depicted in Figure 6 and termed the ‘observed’ overstrength factor. Quantification of the actual Copyright  2002 John Wiley & Sons, Ltd.

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Figure 7. Comparison between the capacity envelopes for the three groups of buildings, obtained from inelastic static pushover analysis; Vy, actual strength (base shear); W, weight of building; D, top displacement; H, height of building; 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

overstrength can be employed to reduce the forces used in the design, hence leading to more economical 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 difference between the actual and the design material strength; (2) conservatism of the design procedure and ductility requirements; (3) load factors and multiple load cases; (4) accidental torsion consideration; (5) serviceability limit state provisions; (6) participation of nonstructural elements; (7) effect of structural 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 the design forces. Copyright  2002 John Wiley & Sons, Ltd.

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Figure 8. Observed ( d) and inherent ( i) overstrength for the 12 buildings; Vy, actual strength; Ve, elastic strength; R, response modification factor; 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 are estimated for the 12 buildings by means of inelastic static pushover as well as incremental time-history analyses up to collapse. The capacity envelopes of the buildings obtained from time-history collapse analyses, which are presented elsewhere (Mwafy and Elnashai, 2000), are utilized to evaluate overstrength factors. The envelopes were developed using regression analysis of the maximum response points (the roof displacement and the base shear) of eight seismic excitations for each building. Figure 7 shows the capacity envelopes for the three groups of building obtained from inelastic pushover analyses using a triangular lateral load distribution, where the base shear and the top displacement are normalized by the weight and the height of the building, respectively. The observed overstrength factors from inelastic static pushover and time-history collapse analyses are depicted in Figure 8(a). The overstrength obtained from an alternative definition, as discussed hereafter, is shown in Figure 8(b). It is noteworthy that the maximum strength of the 12 buildings is generally observed at or before the interstorey drift collapse limit state, with the exception of few cases. In those cases the lateral 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 envelopes shown 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 for the irregular buildings. This is a result of the extra tensile forces imposed on these columns as a result of the cut-off at the ground storey. This is clear in Figure 9, where the progress of hinge formation for a sample building from each group is shown at the global yield limit state. The first observed 10 plastic hinges can be distinguished in the figure from other subsequently formed hinges. The advantages of employing regular structural systems in the design are clear in Figures 7 and 9. The plastic hinges observed in the second group of buildings up to the global yield limit state are mainly in beams. In Figure 9(b), only one plastic hinge in columns is observed (hinge 91 from a total of 96 hinges), Copyright  2002 John Wiley & Sons, Ltd.

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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, top drift ratio = 1
0%); (c) FW-M030 (top displacement = 255 mm, top drift ratio = 1
1%); 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 Copyright  2002 John Wiley & Sons, Ltd.

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Table 4. Comparison of force reduction factor (R) and capacity (Vy) of buildings designed for equal seismic intensity and different ductility levels 0
30g pair

0
15g pair

reduction in R (%)

increase in Vy (%)

reduction in R (%)

increase in Vy (%)

25 25 25

25 8 26

33 33 33

26 34 26

IF-group RF-group FW-group

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

whereas several column and wall hinging cases are observed at this moderate level of lateral load in the irregular frame and the frame-wall structures. For the third group, first yield in vertical elements is observed in the ground storey walls shortly after the first beam yielding. As explained earlier, the stiffness 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 the ground level are generated as a result. It is observed that more plastic hinges are formed in the external frames compared with the internal frames, particularly for the first two groups. For the frame structures, this is attributed to the higher stiffness of the short span external beams, which attract higher forces. The design of the internal frame beams 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 internal beams, 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/W ratios are observed for the 8-storey frame-wall structures, reflecting the high stiffness and the efficiency of this structural system in resisting lateral forces. The lowest V/W values are observed for the 12-storey regular frame buildings as a result of the high total gravity load. However, the maximum base shear of the latter ranges from 7300 kN to 13200 kN, slightly higher than the maximum base shear observed for the 8-storey irregular frame buildings (6600–12700 kN). The severance of four perimeter columns and increasing the height of the ground storey leads to the observed reduction in the lateral stiffness of the irregular buildings. Results for the pairs of buildings designed to the same seismic intensity and different levels of ductility show that the capacity is proportional to the code-defined force reduction factor. Table 4 confirms that the increase in the capacity of lower ductility level buildings compared with their higher ductility level counterparts is consistent with the reduction in the R factors adopted in the design. The only exception is the difference between the 0
30g design ground acceleration pair of the RF-group. It is worth mentioning that in the three groups of buildings equal cross-sectional dimensions were utilized with the buildings designed to the same seismic intensity with the exception of this pair, where the 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 the beams). It is concluded that a direct relationship between the lateral capacity and the design seismic forces 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 Copyright  2002 John Wiley & Sons, Ltd.

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Table 5. Comparison between the design forces and the lateral capacity of buildings designed with an equal force reduction factor and different seismic forces (medium ductility buildings) Reduction in design force level (%)

Reduction in capacity (%)

50 50 50

48 44 52

IF-group RF-group FW-group

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 M015 buildings is almost half that of the M030 buildings (the highest and the lowest envelope). This is summarized in Table 5. It is important to point out that this observation cannot be generalized since cross-sectional dimensions are not equal for buildings designed for different seismic intensity. This is clear 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 of H030 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-sectional dimensions of the H030 buildings. The aforementioned observations show clearly the trade-off between strength and ductility, through the force reduction factor. In another study, Mitchell and Paulter (1994) have commented that it is a common misconception that a structure designed with a force reduction factor R = 4
0 would have half the capacity of another designed with R = 2
0. It is noted in the aforementioned study that member cross-sectional sizes and longitudinal 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 those buildings (4
6 and 3
5) compared with lower ductility level buildings (2
14 and 2
78). This may be attributed to particular requirements of the Canadian design code (CCBFC, 1995). However, the direct relationship between the lateral capacity and the force reduction factor is explicitly confirmed in the present 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 overstrength obtained from dynamic-to-collapse analyses are higher than those obtained from inelastic pushover analyses. This is a result of the conservatism of the design code lateral load distribution in predicting the strength capacity of buildings (Mwafy and Elnashai, 2000). All the 12 buildings studied have overstrength factors over 2
0. The group-3 buildings exhibit the highest level of overstrength, the results of the group-1 and group-2 buildings being comparable, particularly the factors obtained from pushover analyses. For the two buildings designed to the same ductility level in each group, the one designed to a lower seismic intensity exhibits higher overstrength, reflecting the higher contribution of gravity loads. Higher ductility level buildings display higher reserve strength. This is attributed to the use of a higher force reduction factor with these buildings, which causes a reduction in the design forces thus magnifying the effect of gravity loads. The rigorous provisions imposed on these buildings to enhance ductility also lead to increased overstrength. Clearly, the observed reserve strength of the 12 buildings is high, confirming the conservatism of EC8. The actual overstrength factors are expected to be higher than the estimated values from static pushover or even time-history collapse analysis because of to the beneficial effects of some parameters that contribute to overstrength such as nonstructural elements. Copyright  2002 John Wiley & Sons, Ltd.

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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 strength level is to avoid explicit nonlinear structural analyses in the design and allow utilization of elastic analysis methods (CEN, 1994; Uang 1991). It is observed in Figure 7 that the response does not show any significant deviation from the elastic behaviour at first yielding. Hence, this level can be safely reduced to the design level. The aforementioned observation reflects the high reserve strength of EC8designed structures. Studies carried out on buildings designed to US seismic codes have indicated that the overstrength factor varied widely depending on the height of the building, the design seismic intensity and the structural system. A brief review of these studies can be found in Jain and Navin (1995) and in Whittaker, Hart and Rojahn (1999). The scatter of the overstrength results was high, ranging from 1
8 to 6
5 for long and short period range structures, respectively. Some extreme values are also reported in 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 and Vidie (1994) have studied 12 RC buildings to derive overstrength spectra. This has been proposed as a function of the period and the ductility level. The effect of the design intensity and the structural system in the determination of the reserve strength has not been accounted for. Overstrength factors varied 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 than the factors determined from the present investigation by about 30%. This is mainly attributed to the predefined drift limit (1% of the building height) at which the strength of the buildings is calculated from pushover analysis in Fischinger and co-workers’ study. It is clear in Figure 7 that the maximum strength for the currently investigated buildings is observed at higher drift than the limit employed by Fischinger, Fajfar and Vidie (1994). No strength degradation was considered in the analytical models utilized in their study. Therefore, overstrength was calculated at this conservative drift limit because of the uncertainty involved in identifying the maximum strength. Panagiotakos and Fardis (1998) investigated three groups of frame structural systems of various heights. 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 level of ground motion, the results of their study are consistent with the results of the current investigation in terms of the high overstrength factors of EC8-designed buildings. Developing overstrength spectra for RC buildings covering different structural systems, design intensity level, heights and ductility levels requires investigating a considerable range of buildings realistically designed and practically detailed to the code considered. As long as the above is not fulfilled, conservative overstrength for different classes of structures may be set. The rationality and the conservatism of the minimum overstrength factor of 2
0 observed for the investigated sample of medium-rise RC buildings is supported by the following points: . The average contribution to overstrength from the difference between mean and characteristic values of material strength exceeds 1
5. Employing the elastic period in the design adds to overstrength factors by 1
19–1
43, according to the results shown in Figure 5. When designing under unidirectional seismic excitation, EC8 imposes an extra overstrength factor of 1
3 on columns since they are designed under biaxial bending. Consequently, the total overstrength from the above parameters is in excess of 2
0. . The studies carried out on buildings designed to the US codes indicated a minimum overstrength of Copyright  2002 John Wiley & Sons, Ltd.

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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 US codes 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 a result of conservatism of the code lateral load distribution in estimating the capacity and contribution of nonstructural elements, as explained earlier. It is also important to note that previous studies have confirmed that low-rise buildings exhibit higher overstrength compared with medium-rise buildings. Therefore the minimum overstrength of 2
0 can also be applied to this class of building. Moreover, seismic forces generally play a less important role in the determination of cross-sectional sizes and reinforcements than do gravity loads, which govern the design of those buildings. Hence a precise evaluation of overstrength for the purpose of assessing force reduction factors of short period buildings is not of great benefit. Finally, concerning long period RC buildings, it is noteworthy that the overstrength of those buildings might be higher than the values observed for medium-rise buildings since the design is likely to be governed by stiffness (or storey drift; 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 particular important features of seismic response. For instance, it is expected to observe higher overstrength for the 8-storey irregular frame buildings compared with the 12-storey structures. The design code is more stringent for irregular structures and gravity loads play a more significant role in the design of lower rise buildings. This causes relatively higher section dimensions and reinforcement ratios than the demand imposed by seismic forces; hence higher overstrength is anticipated. For two buildings designed to the same seismic intensity (elastic force, Ve) and different ductility levels, each structure will be assigned a different design force level (Vd) as a result of employing a different force reduction factor (R). Thus, Vd1 ˆ

Ve R1

…2†

Vd2 ˆ

Ve R2

…3†

and

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

d1 ˆ

Vy1 Vd1

…4†

d2 ˆ

Vy2 Vd2

…5†

and

The generic definition of overstrength ( d) is the ratio between actual (Vy) and design (Vd) strength. An Copyright  2002 John Wiley & Sons, Ltd.

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

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

i ˆ

Vy d ˆ Ve R

…6†

This definition, which is the inverse of the ductility part of the force reduction factor (R ) as shown in Figure 6, utilizes Ve. Hence the force reduction factor, which is defined on as empirical basis, is avoided in this definition. The suggested measure of response ( i) reflects the reserve strength and the anticipated behaviour of the structure under the design earthquake, as depicted in Figure 10. It is termed as ‘inherent’ to distinguish it from the ‘observed’ overstrength commonly used in the literature. Clearly, in the case of Oi  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 postelastic range. 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 high for the third group of buildings. The strength levels of the four buildings of this group exceed the elastic 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, where minimum section sizes and reinforcements lead to an elastic response for this class of structure under the design intensity. Moreover, it is clear that the response of the buildings designed to a low ductility level in each group are likely to be elastic, which again reflects the conservatism of the code. It is worth mentioning that for such a type of structure no capacity design rules are applied, although some requirements to enhance the ductility are imposed. The lowest force reduction factors of EC8 are assigned to this class of building (1
5–2
5). Even though the buildings designed to this ductility level are 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 not been exploited by the design code to reduce the design forces through employing higher force reduction factors. Contrary to the conventional definition of overstrength ( d), the results for i display clearly the expected higher overstrength of the IF group of buildings compared with the RF group, as shown in Copyright  2002 John Wiley & Sons, Ltd.

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Figure 8. However, for the buildings designed to the same seismic intensity in each group, the higher ductility level buildings show lower values of i, reflecting the higher ratio of the forces that are imposed on the structure in the postelastic range. Finally, the values of i are consistent with the results of the overstrength ( d) in terms of the observed higher values for the buildings designed to lower seismic intensity. 5.

CONCLUSIONS

In this paper, 12 RC buildings with various characteristics were studied to evaluate the overstrength and investigate its relationship with the force reduction factor. The study was carried out by means of eigenvalue, inelastic pushover and time-history collapse analysis employing eight natural and artificial records. Response criteria at both member and storey levels were defined. The input accelerograms were scaled to equal velocity spectrum intensity. Notwithstanding the limitations of using a finite set of input motions and specific structures, the following conclusions are applicable to a large class of RC building. . A conservative prediction of the elongated period of structures can be obtained by employing a reduced flexural stiffness in eigenvalue analysis. Using the effective stiffness values suggested by FEMA 273 (FEMA, 1997) and Paulay and Priestley (1992) in eigenvalue analysis underestimates the inelastic periods. Employing the values suggested by Paulay and Priestley for walls and half of those values for beams and columns show a reasonably conservative prediction of the cracked period at the design intensity. Utilizing this period in the design instead of the elastic period leads to a reduction 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 inelastic periods 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 of the dominant role of gravity loads. Hybrid structural systems and buildings designed to high ductility levels also exhibit high overstrength. This is a result of minimum cross-section—sizes and reinforcements, which are more stringently applied to such buildings. . The contribution to overstrength from three sources exceeds a factor of 2
0 These are: (1) difference between mean and characteristic values of material strength, (2) employing the elastic period in the design instead of the cracked period and (3) designing the columns in biaxial bending when analysed under unidirectional seismic excitation. . Overstrength during earthquakes should be higher than the values obtained from inelastic static analyses using the code lateral load distribution. The triangular load is conservative in predicting the ultimate capacity. Also, contributions of nonstructural elements should produce higher capacity and hence higher overstrength. . If overstrength is not accurately evaluated by means of inelastic analysis, a lower bound may be utilized. A conservative overstrength factor of 2
0 is suggested for medium period RC buildings designed and detailed to EC8 (in principle, this applies to other modern codes). This limit can be applied to low-rise buildings since they usually possess higher overstrength than do medium-rise buildings. . It is suggested to utilize an additional measure of response alongside the widely utilized overstrength measure ( d). This is herein termed the ‘inherent overstrength factor’ i. The former factor relates the ultimate strength to the force assumed in the design, whereas the proposed measure relates the ultimate strength to the elastic force. This avoids implicating the force reduction factor in the definition. The overstrength measure is needed for evaluation and possible calibration of the force Copyright  2002 John Wiley & Sons, Ltd.

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reduction factor adopted in the design. However, it may lead to unreliable predictions of overstrength because of the inclusion of the force reduction factor assumed in the design in its definition. It also failed to confirm clearly the conservatism of the code since the range of its variation is too wide. The suggested measure ( i) better reflects the anticipated behaviour of the structure and the reserve strength under the design earthquake owing to its explicit elastic response limit. REFERENCES

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