Steel Space Frames

February 11, 2004 15:47 WSPC/124-JEE 00136

Journal of Earthquake Engineering, Vol. 8, No. 3 (2004) 457496c Imperial College Press

SEISMIC BEHAVIOUR OF PERIMETER AND

SPATIAL STEEL FRAMES

ELENA MELE, LUIGI DI SARNO and ANTONELLO DE LUCA

Department of Structural Analysis and Design,

University of Naples Federico II,

P.le V.Tecchio, 80,

80125 Naples, Italy

Received 6 May 2002Revised 26 August 2002Accepted 11 June 2003

The present study deals with the seismic performance of partial perimeter and spatialmoment resisting frames (MRFs) for low-to-medium rise buildings. It seeks to establishperimeter configuration systems and hence the lack of redundancy can detrimentallyaffect the seismic response of framed buildings. The paper tackles this key issue by com-paring the performance of a set of perimeter and spatial MRFs, which were consistentlydesigned. The starting point is the set of low- (three-storey) and medium-rise (nine-storey) perimeter frames designed within the SAC Steel Project for the Los Angeles,Seattle and Boston seismic zones. Extensive design analyses (static and multi-modal)of the perimeter frame buildings and consistent design of spatial frame systems, as analternative to the perimeter configuration, were conducted within this analytical study.The objectives of the consistent design are two-fold, i.e. obtaining fundamental periodssimilar to those of the perimeter frames, i.e. same lateral stiffness under design horizon-tal loads, and supplying similar yield strength. The seismic behaviour of perimeter andspatial configuration structures was evaluated by means of push-over non-linear staticanalyses and inelastic dynamic analyses (non linear time histories). Comparisons be-tween analysis results were developed in a well defined framework since a clear schemeto define and evaluate relevant limit states is suggested. The failure modes, either localor global, were computed and correlated to design choices, particularly those concerningthe strength requirements (column overstrength factors) and stiffness (elastic stabilityindexes). The inelastic response exhibited by the sample MRFs under severe groundmotions was assessed in a detailed fashion. Conclusions are drawn in terms of localand global performance, namely global and inter-storey drifts, beam and column plas-tic rotations, hysteretic energy. The finding is that the seismic response of perimeterand spatial MRFs is fairly similar. Therefore, an equivalent behaviour between the twoconfigurations can be obtained if the design is consistent.

Keywords: Seismic design; steel frames; perimeter; spatial; pushover; dynamic analysis.

1. Introduction

The design criteria to obtain satisfactory seismic performance of steel moment re-

sisting frame (MRF) buildings is a major research issue which has been deeply

discussed during recent years (e.g. Luco and Cornell [2000], Yun et al. [2002],

among others). Building damage survey carried out in the aftermath of the 1994

457


458 E. Mele, L. Di Sarno & A. De Luca

Northridge (California) and 1995 Hyogoken-Nanbu (Japan) earthquakes showed

clearly widespread and unexpected damage modes in steel structures. Several

types of brittle fractures were, indeed, found at welded connections in MRFs.

As a consequence, a great deal of research in USA, Japan and Europe has

been devoted to understand and prevent the causes of the observed poor per-

formance of steel structures. For example, in the USA, the SAC Joint Venture

was formed with the specific goal of investigating the damage to welded steel

moment frame buildings in the 1994 Northridge earthquake and developing re-

pair techniques and new design approaches to minimise damage to steel moment

frame buildings in future earthquakes [SAC, 1995]. In the first part of the research

program (SAC Steel Project) interim design guidelines [FEMA 267, 1995; FEMA

267A, 1997; FEMA 267B, 1999] were developed and two sets of nine frame build-

ings (low-, medium- and high-rise) were designed in compliance with both pre-

Northridge design practice [UBC, 1994] and the Interim Guidelines. The structural

systems of these nine model buildings are perimeter MRFs, designed for three dif-

ferent earthquake prone zones. Such structures were considered as benchmarks in

order to carry out analytical investigations on the seismic performance, with the

aim of:

Developing concepts and information needed to identify and predict local and

global seismic demand parameters which have a significant effect on the seismic

response of steel moment resisting frames;

Identifying the effect on seismic performance of the strength, stiffness and

hysteretic characteristics of realistic alternative connection types as well as of

the system configuration and proportioning.

In the present study, the SAC low- and medium-rise buildings were selected and

assessed to establish the governing parameters in the seismic design and hence eval-

uate the response of MRFs under severe earthquakes, varying the design criteria

and the design parameter values. This paper focuses, however, on the influence of

the configuration layout and redundancy of the lateral resisting system on the seis-

mic performance of framed buildings. The SAC perimeter structures are analysed

and contrasted to alternative structural configurations, i.e. spatial MRFs.

2. Object and Layout of the Paper

This analytical study deals with the seismic performance of partial perimeter and

spatial MRFs for low-to-medium rise buildings. Its aim is to establish if the de-

sign choice of employing perimeter configuration system, namely the lack of high

redundancy, has detrimental effects on the seismic response of framed buildings.

The effect of redundant seismic framing is a major topic, often appointed by the

engineering community as one of the possible causes of the poor seismic performance

[FEMA 350, 2000; Whittaker et al., 1999]. Nevertheless, few studies have been

conducted so far to quantify systematically such an effect.


Seismic Behaviour of Perimeter and Spatial Steel Frames 459

The framework employed within this study is based upon the comparison of

the seismic performance of perimeter and spatial MRFs, which are consistently

designed. Thus, the set of low- (three-storey) and medium-rise (nine-storey)

perimeter frames designed within the SAC Steel Project for the Los Angeles, Seattle

and Boston seismic zones was considered as a benchmark. The scheme of the study

carried out can be summarised as follows:

Definition of the design parameters for steel MRF systems. A critical assessment

of MRF design procedures and approaches is carried out in order to establish the

structural parameters governing the seismic design.

Elastic analyses (static and multi-modal) of the SAC perimeter frame buildings

under design load distributions and evaluation of major design parameters.

Consistent design of spatial frame systems, as alternative structural configuration

for the SAC model buildings. The design is consistent because it is carried out

with two specific objectives: (i) obtaining the same period as the perimeter frame

building, i.e. same lateral stiffness under design horizontal loads; (ii) supplying

the same yield strength as perimeter frame building. It is also worth mentioning

that the total weight of both configurations does not differ significantly.

Assessment of the inelastic behaviour of perimeter and spatial frames and

comparisons.

Evaluation of the seismic performance and inelastic demands under severe

earthquakes of perimeter and spatial frames and comparisons.

The seismic behaviour of perimeter and spatial configuration structures was

evaluated by means of push-over non-linear static analyses and inelastic dynamic

analyses. Comparisons between analysis results were developed in a well defined

framework. A clear scheme to define and evaluate relevant limit states (LSs) is

suggested in agreement with a similar proposal available in the literature [Broderick

and Elnashai, 1996; Elnashai et al., 1998]. Thus, the failure modes, either local or

global, were computed and correlated to design choices, particularly concerning

strength requirements (column overstrength factors) and stiffness (elastic stability

indexes). Moreover, the inelastic response exhibited by perimeter and spatial MRFs

under severe ground motions was assessed. Conclusions are drawn in terms of local

and global performance, namely (i) global and inter-storey drifts, (ii) beam and

column plastic rotations and (iii) hysteretic energy.

3. Structural Systems

3.1. Description

The assessed structures are three- and nine-storey office buildings, designed

according to the UBC 1994 provisions for three different seismic zones, respectively

Los Angeles (LA), seismic zone 4, Seattle (SE), seismic zone 3 and Boston (BO),

seismic zone 2A. They were designed by three U.S. professional firms, i.e. Brandow



and Johnston Associates (three- and nine-storey LA buildings), KPFF Consulting

Engineers (three- and nine-storey SE buildings) and LeMessurier Consultants

(three- and nine-storey BO buildings). These MRFs have been extensively anal-

ysed by Gupta and Krawinkler [1998a, 1998b, 2000a, 2000b] within SAC studies on

structural system performance, and other Researchers [e.g. Maison and Bonowitz,

1999; Mele et al., 2000].

Low-rise buildings have a rectangular floor plan 56.08 37.80 m (Fig. 1). The

structural configuration consists of six 9.15 m bays along the major side and four

9.15 m bays in the orthogonal direction. The inter-storey height is 3.96 m. The

lateral load resisting system consists of two four-bays perimeter moment frames in

both principal directions, thus providing torsional resistance, and interior gravity

frames with simple shear connections. In the LA perimeter frame there are only

three moment resisting bays; the fourth bay includes pinned beams connected to

weak axis columns. At the first and second floor, beam span loads (dead and live)

are equal to 14.88 kN/m and joint vertical loads are equal to 158 kN and 107 kN

at interior and perimeter joints, respectively. Beam loads of 12.65 kN/m and joint

vertical loads of 140 kN (interior joints) and 92 kN (exterior joints) were respectively

considered at the roof. At each storey level the slab consists of 76.2 mm metal

decking with 63.5 mm of normal weight concrete (NWC). The latter is stiff enough

to ensure the diaphragm (rigid) action. The total seismic load of the building is

Wtot = 29 500 kN.

The nine-storey buildings (Fig. 2) have a square floor plan 45.7 45.7 m, with

five 9.15 m bays along both sides. The inter-storey height is 5.49 m at the first

storey and 3.96 m at all other storeys, whilst the basement level is 3.66 m in height.

Four MRFs are located along the perimeter of the buildings, two for both principal

directions to provide lateral and torsion resistance, similarly to low-rise frames.

Perimeter systems consist of four bays with strong axis columns rigidly connected

to the girders plus an external bay where the girder is shear connected to the corner

columns, thus biaxial bending is prevented in all building corner columns. Beam

span loads and joint vertical loads are similar to the three-storey counterparts. The

slab is a floor system (rigid plane) of composite metal deck (76.2 mm thick), with

63.5 mm of NWC fill. The total seismic load of the building is Wtot = 90 140 kN.

Element cross sections of perimeter and spatial frames employed in three- and

nine-storey buildings are provided in Table 1 and Table 6, respectively.

Nominal yield strength equal to 345 MPa (50 ksi) was used for girders and

columns of all perimeter frames, except for LA frames, in which nominal yield

strength equal to 248 MPa (36 ksi) was adopted for girders. Similar yield stress

values were employed either for low- or medium-rise, spatial MRFs.

In the following, low- and medium-rise perimeter frames are appointed as:

LA-3P, SE-3P, BO-3P and LA-9P, SE-9P, BO-9P. Similarly, the spatial frames

are appointed as: LA-3S, SE-3S, BO-3S and LA-9S, SE-9S, BO-9S. For example,

LA-3P means that the frame is located in Los Angeles, it has three-storeys and

perimeter configuration.



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Fig. 2. Geometry of nine-storey MRFs: plan layout (left) and vertical section (right).

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Table 1. Beam and column cross sections (three- and nine-storey perimeter MRFs).

Beams Internal Columns External ColumnsFrame

Storey LA SE BO LA SE BO LA SE BO

01 W33 118 W24 76 W21 62 W14 311 W14 176 W14 99 W14 257 W14 159 W14 74

Three-Storey 12 W30 116 W24 84 W21 57 W14 311 W14 176 W14 99 W14 257 W14 159 W14 74

23 W24 68 W18 40 W18 35 W14 311 W14 176 W14 99 W14 257 W14 159 W14 74

01 W36 160 W30 108 W24 68 W14 500 W24 229 W14 233 W14 370 W24 229 W14 159

12 W36 160 W30 108 W36 135 W14 500 W24 229 W14 233 W14 370 W24 229 W14 159

23 W36 160 W30 116 W33 118 W14 455 W24 229 W14 233 W14 370 W24 229 W14 159

34 W36 135 W30 108 W30 116 W14 455 W24 229 W14 233 W14 370 W24 229 W14 159

Nine-Storey 45 W36 135 W27 94 W30 116 W14 370 W24 207 W14 211 W14 283 W24 207 W14 132

56 W36 135 W27 94 W30 108 W14 370 W24 207 W14 211 W14 283 W24 207 W14 132

67 W36 135 W24 76 W30 99 W14 283 W24 162 W14 176 W14 257 W24 162 W14 99

78 W30 99 W24 76 W27 94 W14 283 W24 162 W14 176 W14 257 W24 162 W14 99

89 W27 84 W24 62 W24 76 W14 257 W24 131 W14 120 W14 233 W24 131 W14 68

910 W24 68 W24 62 W18 40 W14 257 W24 131 W14 120 W14 233 W24 131 W14 68



3.2. Modelling

The structural analysis was performed through two finite element method codes.

Elastic analyses were carried out by means of SAP2000 [SAP2000, 1999], while

the computer code DRAIN-2DX [Prakash et al., 1993] was used to evaluate the

responses of both perimeter and spatial MRFs beyond the elastic limit (inelastic

static and non linear dynamic analyses). Moreover, DRAIN-2DX models were cali-

brated on the basis of the SAP2000 counterparts. Bare frames, employing centreline

dimensions, were modelled as two-dimensional assemblages of plastic hinge (lumped

plasticity) beam and beam-column elements. The latter account for bending-axial

interaction, through a linearised biaxial plastic domain (bending moment-axial

load), characterised by horizontal and vertical straight lines corresponding to 10% of

adimensionalised bending and axial capacity, with reference to full plastic bending

moment and squash load, respectively. Bilinear elastic plastic behaviour with strain

hardening of 1% was adopted to model plastic hinges. Shear behaviour of beams and

columns was assumed to remain linearly elastic. Geometric non-linearities, i.e. P

effects, were included within elastic and inelastic analyses. Panel zone yielding and

deformation were not considered. Moreover, the additional stiffness and strength

due to the presence of the concrete deck slab were not accounted for in the analysed

plane systems. Dynamic elastic analyses were based upon a 5% equivalent viscous

damping, whilst the latter was not accounted for in the inelastic analyses since

dissipative sources other than material plasticity were not considered.

4. Design Analyses of Perimeter Frames

The design choices adopted for perimeter framed buildings were investigated with

respect to the main structural properties, namely stiffness (fundamental period and

lateral stiffness), strength (design stress level, column over-strength factor, yield

strength) and stability (sensitivity to P effects). These analyses were based upon

UBC94 seismic provisions [UBC, 1994], which defines the seismic base shear V as

follows:

V = C(T ) W , (1)

where the seismic coefficient (C) is a function of the period (T ):

C =1.25ZIS

RwT2

3

, (2)

where Z = zone factor, I = importance factor, S = soil response factor, Rw =

response modification or force reduction factor.

4.1. Period and design forces

The design period at which the static lateral forces are computed deserves particu-

lar attention and has been subjected to several adjustments in the Codes of Practice



during recent years [Chopra and Goel, 2000]. However, it is common among pro-

fessionals to adopt empirical formulas like that proposed by UBC (Method A):

TA = Ct (Htot)3

4 (3)

(with Ct = 0.035 for steel frames and Htot is the global height in feet), albeit

the code specifically advises the adoption of a more refined, Rayleigh type formula

(Method B). In such case, as an upper limit of the period to be used in the as-

sessment of structural deformations, the code requires, for high seismicity areas

(Zone 4):

TB,max = 1.30 TA , (4)

whilst a longer value, i.e.

TB,max = 1.40 TA (5)

is allowed for low-to-intermediate seismicity, namely Zones 1, 2 and 3.

For the assessed buildings Methods A and B were used to determine the design

base shears. The period values computed by means of Eq. (3) and either Eqs. (4)

or (5), depending on seismic zone, along with the actual values of the structure

fundamental periods, are provided in Table 2 for three- and nine-storey perimeter

MRFs. In these tables the design base shear (VUBC-A and VUBC-B) and maximum

inter-storey and roof drift ratios (d/h and Dtop/H), computed by means of elastic

analyses, are also given. For the sake of completeness, base shears corresponding to

wind forces and the relative deformation parameters were evaluated; the latter are

included in the above mentioned table. To evaluate wind actions and effects, wind

speeds of 70 mph (LA) and 80 mph (BO and SE) and Exposure B were assumed.

Large scatters can be observed between the actual natural periods and the

counterparts based upon code equations. Standardised semi-empirical formulae are

calibrated to account for the contribution of in-filled non structural elements. The

latter increase the frame lateral stiffness. These contributions are, however, usually

neglected in preliminary design models. For the set of SAC buildings, elastic anal-

yses performed on 3D FEM frame models showed that, due to the contribution of

the internal frames (gravity load system), the period values decrease by less than

5% for the three-storey buildings and by approximately 10% for the nine-storey

buildings. Thus, unless very stiff non-structural elements are adopted, empirical

code formulas provide too low an estimate of the fundamental period. This finding

is in agreement with previous studies [Gupta and Krawinkler, 1998b, 2000a].

4.2. Lateral stiffness

Elastic deformation parameters, i.e. inter-storey and global drifts, were computed

through static analyses employing Methods A and B (UBC, 1994) horizontal force

distributions, as well as through response spectrum analysis (Table 2).

It is instructive to note that in steel MRF design, stiffness requirements, ex-

pressed as either maximum inter-storey or roof drift limitations, are usually more

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Table 2. Fundamental period, base shear, maximum drift index and roof drift (three- and nine-storey perimeter MRFs).

Method A Method B Wind Loads Dynamic Analysis

T V/W d/h Dtop/H T V/W d/h Dtop/H V/W d/h Dtop/H T Dtop/H

Frame [sec] [%] [%] [%] [sec] [%] [%] [%] [%] [%] [%] [sec] [%]

LA-3P 7.64 0.30 0.26 0.72 6.26 0.25 0.22 2.36 0.07 0.06 0.99 0.14

SE-3P 0.55 5.73 0.38 0.33 0.77 4.48 0.30 0.26 3.08 0.15 0.13 1.31 0.09

BO-3P 2.87 0.35 0.30 0.77 2.24 0.29 0.24 3.08 0.27 0.23 1.76 0.09

LA-9P 4.04 0.30 0.28 1.79 3.39 0.25 0.24 2.53 0.18 0.13 2.53 0.12

SE-9P 1.38 3.04 0.41 0.36 1.93 2.42 0.33 0.29 3.32 0.35 0.27 3.30 0.10

BO-9P 1.51 0.28 0.21 1.93 1.21 0.22 0.18 3.32 0.59 0.32 3.63 0.10



stringent than strength. For code compliant design [UBC, 1994] the inter-storey

drift limitation is equal to either 1/300 or 1/400 for structures with fundamen-

tal period respectively either lower or higher than 0.7 sec. Thus, for three-storey

buildings, 1/300 applies when Method A is used, since Eq. (3) provides a period

value equal to 0.55 sec; on the other hand, 1/400 applies if Method B or spectrum

analysis are adopted.

Inter-storey drift ratios computed for low-rise frames show that code limits

are exactly fulfilled only in LA-3P, while they are slightly exceeded in SE-3P and

BO-3P. However, for such buildings, the roof drift is respectively less than 1/300

and 1/400 under Methods A and B design loads. Roof drift values obtained through

response spectrum analyses are even smaller than the latter ones.

It should be pointed out that for the BO-3P building the base shear due to wind

actions is larger than the seismic shear. Nevertheless, the maximum deformations

occur under the seismic force distribution, due to a larger overturning moment.

Thus, it is likely that member proportioning was governed by the seismic load

condition.

With regard to the medium-rise buildings, the design load due to wind actions

is higher than the seismic ones for BO-9P and SE-9P. It is interesting to note

that whereas for BO-9P the effects due to wind actions (d/h and Dtop/H) are

also greater than the seismic loads, for the SE-9P building the roof and inter-storey

drifts induced by wind actions are slightly smaller than those induced by earthquake

loading.

The LA-9P building fully satisfies the drift limitation (1/400) under Method B

design loads, while SE-9P and BO-9P slightly exceed it. However, the nine-storey

buildings spectrum analysis provides values of the deformation parameters which

are also by and large within code limitations.

Concerning stiffness distribution, the structures exhibit a quite smooth variation

of global stiffness over the height. In the low-rise frames while the beam cross

sections are reduced over the frame height (with the exception of the SE-3P frame,

which has second floor beams stiffer than first storey ones), the columns cross

sections are the same. In the medium-rise frames both columns and beams are

varied throughout the building height. Column splices are generally located at every

third storey, and beams are consistently reduced. In SE-9P, as for SE-3P, second

floor beams are stiffer than first storey ones.

4.3. Design stress level and yield strength

The stress level in structural members under gravity and seismic lateral loads

(Methods A and B of UBC) was evaluated and very low stress values were obtained.

Thus, the base shear Vy corresponding to first yield, i.e. onset of yield strength

in the most stressed section, was evaluated. Values of Vy , normalised to seismic

weight W and to the seismic base shears (VUBC-A and VUBC-B), are provided for

three- and nine-storey buildings in Table 3; the values for wind actions are also

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Table 3. Non-dimensional base shear (three- and nine-storey perimeter MRFs).

VUBC-M.A/W VUBC-M.B/W VWIND/W Vy-triang./W Vy-unif./W Vy-triang./ Vy-triang./ Vy-unif./

Frame [%] [%] [%] [%] [%] VUBC-M.A VUBC-M.B VWIND

LA-3P 7.64 6.26 2.36 20.1 22.79 2.63 3.23 9.66

SE-3P 5.73 4.48 3.08 18.59 21.02 3.24 4.15 6.82

BO-3P 2.87 2.24 3.08 10.89 11.25 3.79 4.86 3.65

LA-9P 4.04 3.39 2.53 10.18 10.61 2.52 3.00 4.19

SE-9P 3.04 2.42 3.32 8.06 8.43 2.65 3.33 2.54

BO-9P 1.51 1.21 3.32 7.53 7.79 4.99 6.22 2.35



included. Design over-strength factors, herein defined as the ratio Vy/VUBC, are

high. They range between 2.52 and 3.25, except where the design was governed by

wind (BO frames). These large values are due to several features: (i) the ratio of

yield-to-allowable stress (about 1.5), (ii) the compliance to drift limitations, (iii)

the compliance of SCWB requirement and (iv) the different susceptibility to second

order effects. The latter is significant for BO MRFs whose Vy/VUBC represent an

outlier when compared to the mean values.

It is worth mentioning that the design of BO-3P as well as SE-9P and

BO-9P, which exhibit over-strengths equal to 3.80, 2.65 and 4.97, respectively, were

governed by wind loading. Thus, the computed Vy/VUBC factors do not refer to the

actual design condition.

Finally, it is interesting to note that for BO frames (both low- and medium-

rise), as well as for SE medium-rise frames, the over-strength factors due to the

wind are lower than the seismic counterparts. By contrast, LA frames (three- and

nine-storeys) and SE low-rise frame, exhibit design over-strength for wind greater

than for earthquake loads.

4.4. Column over-strength factor

Column over-strength factors (COF s) were computed at each frame joint using the

following standardised relationship [UBC, 1994]:

COF =Zc(fyc fa)

Zbfyb(6)

in order to verify the fulfilment of the strong column weak beam (SCWB) re-

quirement (COF 1). In the above equation Zc, Zb and fyc, fyb are section plastic

modulus and yield stress respectively for columns and beams framing into the joint,

while fa is the axial stress in column members. An adequate column over-strength

factor gives rise to a configuration where columns are protected from plastic hinge

formation. Hence, inelastic deformations should occur in beams.

It is worth mentioning that COF s in three-storey frames (Table 4) are greater

than unity, minimum values being 1.06, 1.23 and 1.94, respectively, for BO-3P,

Table 4. COF s (three- and nine-storeyperimeter MRFs).

Frame Minimum Maximum

LA-3P 1.94 (1) 3.77 (3)

SE-3P 1.23 (2) 3.60 (3)

BO-P 1.06 (1) 1.87 (2)

LA-9P 1.39 (7) 4.89 (8)

SE-9P 1.17 (9) 3.66 (1)

BO-9P 0.68 (2) 2.04 (1)

Keys: Bracketed values show the floors.



SE-3P and LA-3P buildings. Moreover, for BO-3P nearly 80% of the total number

of beam-to-column connections exhibit COF s between 1.0 and 1.50, thus show-

ing high risk of extensive column hinging. Indeed, recent studies [Nakashima and

Sawaizumi, 2000] showed that plastic hinges in columns cannot be prevented by

simply satisfying the requirement COF 1. Minimum COF values strongly de-

pend on ground motions and importance of higher vibration modes of vibration. As

rule of thumb, COF s ranging from 1.5 to 2 are required to ensure elastic response of

column members. As a consequence, for low-rise buildings, the adequate fulfilment

of SCWB can be predicted only for LA-3P.

Values of COF vary quite significantly for medium-rise buildings (Table 4) as

a function of seismic hazard. Indeed, for LA-9P, values ranging from 1.4 to 1.9

occur between the fourth and sixth floor, while values greater than 2 can be found

elsewhere. In BO-9P building, COF values smaller than 1 characterise all beam-to-

column joints from the third to seventh floor. Values equal to unity (eighth floor)

or greater can be found in the remaining floors. An intermediate situation can be

observed for SE-9P, exhibiting COF between 1.4 and 1.6 (first and third floor) or

1.7 and 2.1 (fourth and seventh floor); values greater than 2 are located at the

eighth floor. Thus, for LA-9P a SCWB behaviour is expected, while for BO-9P a

weak column strong beam (WCSB) behaviour at intermediate storeys, with col-

umn plastic hinges, seems to be unavoidable. Finally, in SE-9P, plastic hinges are

expected mainly in beam members, although some column plastic deformations are

likely to occur.

4.5. Stability

The design criteria for MRFs are based also on specific checks for second or-

der effects at each storey and overall stability. P effects can be approximately

evaluated through the stability index () expressed as follows:

=Ptot drVtot h

, (7)

where Ptot is the total gravity storey load, Vtot the total seismic storey design shear,

whilst dr and h are, respectively, the design inter-storey drift and relative height.

The stability index expressed by Eq. (7) may be assumed approximately equiv-

alent, for framed structures, to the elastic critical vertical load ratio [Horne, 1975],

defined as the ratio (P/Pcr) in which P is the total vertical load and Pcr is the

elastic critical load value for failure in a sway mode. Therefore, it represents a

measure of the sensitivity to P effects. In International Standards [UBC, 1994;

1997; EC8, 1998], it is required that the ratio should not to exceed 0.10 in order

to neglect P effects. On the other hand, if 0.10 < 0.30, then P effects

should be accounted for; the upper bound for is, however, assumed equal to 0.30.

The -value was computed for all floors in low- and medium-rise frames, under

seismic actions. The maximum values, along with the storey level where they are



Table 5. Stability index (three- and nine-storey perimeter MRFs).

Method A Wind

Frame Minimum Maximum Minimum Maximum

LA-3P 0.017 (3) 0.026 (2) 0.021 (3) 0.026 (2)

SE-3P 0.029 (3) 0.043 (2) 0.038 (3) 0.044 (2)

BO-P 0.049 (3) 0.039 (2) 0.064 (3) 0.080 (2)

LA-9P 0.022 (9) 0.065 (1) 0.046 (7) 0.065 (1)

SE-9P 0.030 (9) 0.096 (1) 0.057 (9) 0.097 (2)

BO-9P 0.049 (9) 0.154 (1) 0.069 (7) 0.146 (1)


found, were also evaluated (Table 5). The values corresponding to wind effects were

included for comparative purposes.

For low-rise MRFs, the stability check is greatest at the second floor, indepen-

dently of seismic region. The maximum values for are always lower than 0.10, with

the highest value for BO-3P frame (max = 0.080), whilst the greatest safety margin

for stability checks is exhibited by LA-3P (max = 0.026). Intermediate behaviour

characterises the SE-3P building (max = 0.044).

For nine-storey buildings the variation of stability index heightwise is more

pronounced than in low-rise frames, with the highest values are at either the

first or second floor, and a nearly linear reduction from the base to roof level.

The most critical stability indexes were found for the BO-9P building (max =

0.154), whilst greater safety margin for stability checks is exhibited by LA-9P frame

(max = 0.065). An intermediate behaviour characterises the SE-9P frame (max =

0.097).

As a result, second-order effects can be significant for BO-9P and, to a lesser

extent, SE-9P. These frames possess the highest values of the stability index; similar

results were found for BO-9S and SE-9S. However, the latter are characterised by

lower values than the perimeter frames, thus rendering the onset of instability

more unlikely.

5. Design of Spatial Frames and Comparison to Perimeter Frames

Spatial MRF buildings possess the same layout as their perimeter counterparts,

see for example Fig. 1 (low-rise) and Fig. 2 (medium-rise). For such structures, it

was assumed that all frames along the North-South direction are special moment

resisting frames. Braced frames are assumed to resist earthquake loads in the

orthogonal direction.

Floor masses for spatial MRFs were computed by considering 1/7 floor tributary

area (low-rise) or 1/6 (medium-rise). The total seismic load of a three-storey

building is Wtot = 29 500 kN; thus for spatial MRF is: WMRF-S = 4214 kN

versus WMRF-P = 14750 kN for the perimeter MRF. For nine-storey buildings



it is Wtot = 90 140 kN, thus for spatial MRF is: WMRF-S = 15023 kN versus

WMRF-P = 45 070 kN for the perimeter MRF.

The design of buildings with spatial frames was carried out according to UBC

seismic provisions [UBC, 1994]. The design target was to match the stiffness

and strength properties of their perimeter counterparts. Member cross sections

employed are provided in Table 6.

The results of comparative analyses for the design parameters of both spatial

and perimeter frames are discussed hereafter.

5.1. Period and design forces

The fundamental periods of low- and medium-rise spatial frames are provided in

Table 7. The values computed by means of code equations are equal to perimeter

values because the building height does not change. Similarly, for the design base

shears, which are based on the spectral ordinates.

Negligible variations of the fundamental period were found for low-to-moderate

seismic zone (BO and SE). Spatial frames are slightly stiffer than perimeter frames;

however, variations do not exceed 8%. Similarly, medium-rise frames show roughly

the same periods for either perimeter or spatial configurations.

Finally, modal masses relative to the first vibration mode were computed for the

assessed buildings; such values were then compared. Two points are worth noting:

(i) all frames possess a modal mass nearly equal to 85% and (ii) modal masses of

perimeter structures are roughly equal to spatial ones; the variations are, in fact,

within 3%. As a result, a dynamic response correspondent to a single degree of

freedom system (SDOF) may be predicted for the set of frames, either three- or

nine-storeys.

5.2. Lateral stiffness

Table 7 provides maximum values of elastic deformation parameters (inter-storey

and roof drift ratios) under design loads for low- and medium-rise spatial structures.

Values relative to low-rise frames are compliant with code limitations, for either

Method A (d/h 1/300) or Method B (d/h 1/400). This is not the case for SE-

3S; however, the exceedence is around about 5% and hence is negligible. By contrast,

the spatial nine-storey frame design gave rise to structures (LA-9S and BO-9S)

exceeding slightly standardised values, namely 1/400; higher d/h were found for SE-

9P. Maximum inter-storey drifts are not found at the same floor, being the second

for BO-9S and fourth for LA-9S. Drift ratios computed for spatial configurations

replicate the response established for perimeter frames.

5.3. Design stress level and yield strength

Base shears Vy corresponding to first yield, normalised to seismic weight (W),

i.e. Vy/W , and to seismic base shears (VUBC-A and VUBC-B), are given in Table 8

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Table 6. Beam and column cross sections (three- and nine-storey spatial MRFs).

Beams Internal Columns External Columns

Frame Storey LA SE BO LA SE BO LA SE BO

01 W18 71 W14 48 W14 26 W14 120 W14 99 W14 61 W14 120 W14 99 W14 61

Three-Storey 12 W18 65 W14 48 W14 26 W14 120 W14 99 W14 61 W14 120 W14 99 W14 61

23 W18 55 W14 26 W14 22 W14 120 W14 99 W14 61 W14 120 W14 99 W14 61

01 W24 104 W18 71 W18 60 W14 283 W14 211 W14 211 W14 283 W14 211 W14 211

12 W24 117 W18 71 W18 60 W14 283 W14 211 W14 211 W14 283 W14 211 W14 211

23 W24 117 W18 71 W18 60 W14 211 W14 211 W14 211 W14 211 W14 211 W14 211

34 W24 84 W18 71 W18 60 W14 211 W14 211 W14 211 W14 211 W14 211 W14 211

Nine-Storey 45 W24 84 W18 76 W18 55 W14 159 W14 193 W14 193 W14 159 W14 193 W14 193

56 W24 84 W18 76 W18 55 W14 159 W14 193 W14 193 W14 159 W14 193 W14 193

67 W24 84 W18 76 W18 50 W14 120 W14 176 W14 176 W14 120 W14 176 W14 176

78 W24 62 W18 71 W18 50 W14 120 W14 176 W14 176 W14 120 W14 176 W14 176

89 W24 62 W18 71 W18 50 W14 90 W14 159 W14 159 W14 90 W14 159 W14 159

910 W24 62 W18 71 W18 50 W14 90 W14 159 W14 159 W14 90 W14 159 W14 159

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E.M

ele,L.D

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eLuca

Table 7. Fundamental period, base shear, maximum drift index and roof drift (three- and nine-storey spatial MRFs).

Method A Method B Wind Loads Dynamic Analysis

T V/W d/h Dtop/H T V/W d/h Dtop/H V/W d/h Dtop/H T Dtop/H

Frame [sec] [%] [%] [%] [sec] [%] [%] [%] [%] [%] [%] [sec] [%]

LA-3S 7.64 0.30 0.25 0.72 6.24 0.25 0.20 2.36 0.07 0.06 0.99 0.15

SE-3S 0.55 5.73 0.35 0.29 0.77 4.48 0.27 0.23 3.08 0.13 0.11 1.22 0.10

BO-3S 2.87 0.32 0.26 0.77 2.24 0.25 0.21 3.08 0.25 0.20 1.63 0.10

LA-9S 4.04 0.29 0.27 1.79 3.39 0.26 0.24 2.53 0.18 0.14 2.52 0.14

SE-9S 1.38 3.04 0.40 0.33 1.93 2.42 0.27 0.23 3.32 0.39 0.27 3.30 0.10

BO-9S 1.51 0.25 0.21 1.93 1.21 0.21 0.17 3.32 0.49 0.34 3.56 0.10

Table 8. Non-dimensional base shear (three- and nine-storey spatial MRFs).

VUBC-M.A/W VUBC-M.B/W VWIND/W Vy-triang./W Vy-unif./W Vy-triang./ Vy-triang./ Vy-unif./

Frame [%] [%] [%] [%] [%] VUBC-M.A VUBC-M.B VWIND

LA-3S 7.64 6.26 2.36 22.94 26.88 3.00 3.68 11.39

SE-3S 5.73 4.48 3.08 22.38 30.23 3.91 5.00 9.81

BO-3S 2.87 2.24 3.08 9.74 11.33 3.39 4.35 3.68

LA-9S 4.04 3.39 2.53 7.32 11.44 1.81 2.16 4.52

SE-9S 3.04 2.42 3.32 10.36 11.08 3.41 4.28 3.34

BO-9S 1.51 1.21 3.32 7.85 8.80 5.20 6.49 2.65



for three- and nine-storey spatial buildings; values due to wind actions are also

provided.

Slightly higher yield thresholds Vy/W were estimated for LA-3S, SE-3S and

SE-9S. By contrast, smaller values arise for BO-3S and LA-9S.

Over-strength factors (Vy/VUBC-A) for the spatial configuration exceeds

perimeter values; this is not the case for BO-3S and LA-9S. Such design

values range between 2.68 and 3.90 (low-rise), 1.81 and 5.20 (medium-rise).

However, referring to the over-strength factor Vy/VWIND , where the design is

governed by wind (SE-9S and BO-9S), the range upper bound is 3.34.

Yield stress level in LA-9S drops off by about 40% with respect to LA-9P, while

SE-9S gives rise to a Vy/VUBC-A that is nearly 30% higher than SE-9P.

As a result, it can be argued that the design of spatial MRFs is governed by

stiffness rather than strength requirements similar to perimeter configurations (see

Sec. 5.2).

5.4. Column over-strength factor

To check SCWB fulfillment, COF s were computed by means of Eq. (7); minimum

and maximum values are given in Table 9 for low-to-medium rise.

Column over-strength factors in spatial frames are greater than unity, the mini-

mum values being between 1.28 and 2.07 and the maximum varying between 4.3 and

8.1. The latter occur at beam-to-external-column joints and can be explained by

the design choice of adopting equal sections for either external or internal columns.

However, three-storey frames exhibit on average higher COF s: in SE-3S these

factors are at least 2.07, so that the occurrence of plastic hinges in beams can be

safely predicted. Moreover, SCWB-type behaviour is likely to occur in LA-3S and

BO-3S. COF s greater than 2 were computed at all beam-to-column connections

in the first two storeys. Therefore, it can be stated that low-storey spatial frames

were designed relying on high COF s in such a way as to guarantee plastification in

beams; for perimeter frames this response can be predicted only for LA.

In medium-rise frames, COF s are greater than 2 for BO-9S, except at the roof

floor (1.38), thus the SCWB design was fulfilled. However, LA-9S and SE-9S are

Table 9. COF s (three- and nine-storeyspatial MRFs).

Frame Minimum Maximum

LA-3S 1.28 (3) 4.29 (2)

SE-3S 2.07 (3) 5.42 (2)

BO-3S 1.76 (3) 5.87 (2)

LA-9S 1.42 (1) 6.16 (8)

SE-9S 1.25 (9) 5.50 (7)

BO-9S 1.38 (9) 5.59 (1)




characterised by intermediate floors where column over-strength drops below 2.

Indeed, LA-9S, between the first and fifth floor, has COF s slightly lower than

2, with minimum equal to 1.42 (Table 9). SE-9S shows values ranging between

1.25 (roof level) and 5.50 (seventh floor); average values are, however, nearly 2.

Maximum values of the column over-strength factor are generally higher than for

the perimeter due to the design choice of employing similar column sections, either

for interior or exterior bays.

5.5. Elastic stability

Stability indexes for spatial frames were evaluated through Eq. (8); the most critical

values and the storey levels at which they occur are given in Table 10.

For three-storey frames, -values were generally found to be more critical for

perimeter rather than spatial configurations. Wind loads give rise to stability in-

dexes as for seismic actions; the criticality is yet to be found at the second storeys.

For BO-3P (Table 5) the maximum (0.080) is computed under wind loads; in

BO-3S the same value ( = 0.069) was evaluated for earthquake and wind.

On the other hand, nine-storey frames exhibit the highest -values (Table 10)

either at ground (LA-3S) or second storeys (SE-9S and BO-9S); lower -values char-

acterise spatial configurations. The variation of stability index throughout the build-

ing height is more pronounced than in low-storey frames: a nearly linear variation,

from ground floor to roof, is observed.

Similarly to low-rise frames, higher -values characterise perimeter rather than

spatial frames. Indeed a, stability index equal to 0.154 was computed for BO-9P

when the seismic force distribution (Method A) is considered. By contrast, in LA-9S,

the sensitivity to second order effects is negligible under both environmental actions,

since = 0.065.

The lowest elastic critical multiplier (1/) for medium-rise building was found

for BO, namely 7.5 (6.5 in BO-9P); whilst a 1/ equal to 16.3 renders the LA

structure safe for stability checks.

Table 10. Stability index (three- and nine-storey spatial MRFs).

Method A Wind

Frame Minimum Maximum Minimum Maximum

LA-3S 0.015 (3) 0.026 (3) 0.020 (1) 0.025 (2)

SE-3S 0.026 (3) 0.039 (2) 0.027 (1) 0.039 (2)

BO-3S 0.044 (3) 0.069 (2) 0.048 (1) 0.069 (2)

LA-9S 0.018 (9) 0.061 (1) 0.042 (8) 0.061 (1)

SE-9S 0.018 (9) 0.098 (2) 0.034 (9) 0.109 (2)

BO-9S 0.029 (9) 0.134 (2) 0.053 (9) 0.123 (2)




Therefore, the computed figures (Table 5 for perimeter and Table 10 for spa-

tial frames), highlight a large increase of susceptibility to second order effects for

medium rather than low rise frames, either perimeter or spatial.

6. Inelastic Static Pushover Analyses

Non linear static push-over analysis is a simplified but powerful tool [Krawinkler

and Seneviratna, 1998; Mwafy and Elnashai, 2000]. It enables the assessment of the

structural behaviour in the elastic and inelastic range and hence the strength and

deformation capacity can be estimated. Therefore, the structural models described

in Sec. 3.1 were implemented and global load-deformation curves were obtained for

perimeter and spatial MRFs.

0

1000

2000

3000

4000

5000

0.00 1.00 2.00 3.00 4.00 5.00 6.00

LA-3P LA-3S UBC (MET.A) YLS ULS

VB (kN)

dtop / Htot (%)

1/400

0

10

20

30

40

50

0.00 1.00 2.00 3.00 4.00 5.00 6.00


VB / W

tot (%)

dtop / Htot (%)

1/400

0

1000

2000

3000

4000

5000

0.00 1.00 2.00 3.00 4.00 5.00 6.00

SE-3P SE-3S UBC (MET.A) YLS ULS

VB (kN)

dtop / Htot (%)

1/400

0

1000

2000

3000

4000

5000

0.00 1.00 2.00 3.00 4.00 5.00 6.00


VB (

kN

)

dtop / Htot (%)

1/4

00

0

10

20

30

40

50

0.00 1.00 2.00 3.00 4.00 5.00 6.00


VB /

Wto

t (%

)

dtop / Htot (%)

1/4

00

0

1000

2000

3000

4000

5000

0.00 1.00 2.00 3.00 4.00 5.00 6.00


VB (

kN

)

dtop / Htot (%)

1/4

00

0

10

20

30

40

50

0.00 1.00 2.00 3.00 4.00 5.00 6.00


VB /

Wto

t (%

)

dtop / Htot (%)

1/4

00

0

1000

2000

3000

4000

5000

0.00 1.00 2.00 3.00 4.00 5.00 6.00

BO-3P BO-3S UBC (MET.A) YLS ULS

VB (

kN

)

dtop / Htot (%)

1/4

00

0

10

20

30

40

50

0.00 1.00 2.00 3.00 4.00 5.00 6.00


VB /

Wto

t (%

)

dtop / Htot (%)

1/4

00

Figure 3. - Push-over curves with triangular distribution (3-story MRFs): single frames (left) and buildings (right).

0

1000

2000

3000

4000

5000

0.00 1.00 2.00 3.00 4.00 5.00 6.00


VB (kN)

dtop / Htot (%)

1/400

0

10

20

30

40

50

0.00 1.00 2.00 3.00 4.00 5.00 6.00


VB / W

tot (%)

dtop / Htot (%)

1/400

Fig. 3. Push-over curves with triangular distribution (three-storey MRFs): single frames (left)and buildings (right).



Figures 3 and 4 provide the results of the analyses carried out on three- and

nine-storey frames, respectively. The results of perimeter and spatial frames, are

contrasted in the same plot. They are expressed in terms of base shear divided

by tributary seismic weight, V/W , versus top displacement normalised to build-

ing height (dtop/Htot). It is deemed useful for comparative purposes to include in

Figs. 3 and 4, plots with curves employing the dimensional base shear V as or-

dinates. Similarly, vertical lines corresponding to the code inter-storey drift limit

[UBC, 1994)], i.e. d/h = 1/400, are also drawn.

In order to assess strength and deformation capacities of MRF systems, two

limit conditions, namely yield limit state (YLS) and ultimate limit state (ULS), were

defined and relevant points are indicated on the curve response. These LS definitions

were chosen consistently with (i) international codes and recommendations [UBC,

0

2000

4000

6000

8000

0.00 1.00 2.00 3.00 4.00 5.00 6.00

LA-9P LA-9S UBC (MET. A) YLS ULS

dtop / Htot (%)

VB (kN)

1/400

0

10

20

30

40

50

0.00 1.00 2.00 3.00 4.00 5.00 6.00

LA-9P LA-9S UBC (MET. A) YLS ULS

dtop / Htot (%)

VB / W

tot (%) 1/400

0

2000

4000

6000

8000

0.00 1.00 2.00 3.00 4.00 5.00 6.00

SE-9P SE-9S UBC (MET. A) YLS ULS

dtop / Htot (%)

VB (kN)

1/400

0

10

20

30

40

50

0.00 1.00 2.00 3.00 4.00 5.00 6.00

SE-9P SE-9S UBC (MET. A) YLS ULS

dtop / Htot (%)

VB / W

tot (%) 1/400

0

2000

4000

6000

8000

0.00 1.00 2.00 3.00 4.00 5.00 6.00

BO-9P BO-9S UBC (MET. A) YLS ULS

dtop / Htot (%)

VB (kN)

1/400

0

10

20

30

40

50

0.00 1.00 2.00 3.00 4.00 5.00 6.00

BO-9P BO-9S UBC (MET. A) YLS ULS

dtop / Htot (%)

VB / W

tot (%) 1/400

Fig. 4. Push-over curves with triangular distribution (nine-storey MRFs): single frames (left)and buildings (right).



1997; EC8, 1998; FEMA 350, 2000] and (ii) experimental and numerical studies

available in the inherent literature [Elnashai et al., 1998; Mwafi and Elnashai, 2000].

Performance criteria used to define the two LSs are briefly described in the following

paragraph; then a detailed discussion of push-over results is provided.

6.1. Yield and ultimate limit states definition

YLS corresponds to the state at which inelastic deformations begin to contribute

to the energy absorption. It can be reliably evaluated by means of static analyses

and corresponds to the onset of the yield at the member far ends.

The ULS definition appears more complex and controversial than YLS, since it

deals with the structure collapse condition, which is always conventionally defined.

In this paper different structural failure modes, both local and global, were con-

sidered, and the corresponding performance criteria were defined. In the following

section, local and global failure sources are described and the corresponding limit

values of relevant performance parameters are provided.

Local failure modes include either member or connection failure. In this paper

only connection failure, due to the attainment of plastic rotation capacity for welded

connections (set equal to SAC target plastic rotation 3%), was accounted for as a

local failure mode. Moreover, base column rotations were also considered to detect

the LSs.

The global failure modes were defined either at the attainment of a collapse

mechanism formation or when a specific threshold value of inter-storey drift,

stability index, lateral resistance reduction, were attained.

The inter-storey drift ratio (IDR) is an adequate structural parameter to assess

the MRF performance. Indeed, it can be checked easily by field evidence in the

aftermath of earthquakes as well as in experimental tests. So far, several researchers

[Leelataviwat et al., 1999, Maison and Bonowitz, 1999] have proposed different IDR

values, as a function of the probability of exceedance and the acceptable damage.

In this paper the IDR value of 4% [FEMA 350, 2000] was assumed as a rational

and reliable limit for structural integrity.

The overall stability under seismic loading implies that the structural system

may preserve its vertical load-bearing capacity at the occurrence of an input ground

motion that provokes both horizontal and vertical vibrations and hence correspon-

dent inertial forces. Indeed, P effects of each storey of the structure can be

evaluated by means of the stability index ; in the present study the limit value of

0.30 was assumed to be the critical threshold for .

The occurrence of both local (i.e. soft storey) and global mechanisms was also

checked. It is worth pointing out that, while a global mechanism failure mode can

occur if there is not a concentration of deformation and strength demand anywhere

in the frame (thus both IDR and -values are less than the limit values), the local

mechanism is likely to be associated with the achievement of the ULSs specified

above, e.g. inter-storey drift and storey stability index.



Finally, a strength degradation criterion, corresponding to a drop of 10% in

the frame lateral resistance, was also considered. It is well known that a global

force-displacement response characterised by a final negative slope occurs when

P effects significantly affect the structural behaviour. Despite the decrease in

the lateral resistance which does not imply failure in itself, the response tends to

be unstable and unreliable, thus the above limit criterion was assumed and checked

for the MRFs.

6.2. Analysis results

Pushover curves provided in Figs. 3 and 4 for low- and medium-rise frames respec-

tively, confirm that frame designs were governed primarily by stiffness requirements.

Indeed, UBC inter-storey drift limit, e.g. 1/400, intercepts the curves at base shears

equal or greater than design values (VUBC). Furthermore, pushover curves provide

information on structural over-strengths, deformation capacity and importance of

P effects.

Values of design, yield and ultimate base shear (respectively VUBC, Vy , Vu) are

depicted on the curves by means of triangular, square, and circular markers, respec-

tively; the same shear values, normalised to the tributary seismic weight, W , are

reported in Table 11 for perimeter and spatial frames, together with the coefficients:

y (design over-strength, equal to the ratio Vy/VUBC), u (global over-strength,

equal to the ratio Vu/VUBC), and u/y (system over-strength).

The ultimate strength, Vu, is the value of base shear corresponding to the struc-

tural collapse, which has been conventionally defined as the minimum of a set of

ULSs, as presented earlier.

For all MRF buildings, particularly the spatial ones, a significant over-strength

above the design base shear VUBC (allowable stress design level) can be observed.

In particular, a global over-strength u, ranging between 3.55 (LA-3P) and 5.81

(SE-3S) for three-storey buildings and between 3.13 (SE-9P) 7.59 (BO-9S) for

nine-storey buildings can be derived from the curves.

Global over-strength, given by the u coefficient, arises from different sources

which can be quantified through the coefficients y and u/y. The contribution

quantified by the y multiplier, which has been appointed as design over-strength,

is partially related to the ratio of yield to allowable stress (about 1.5) and partially

related to the fulfilment of drift limitations as well as to SCWB requirements.

Furthermore, stress redistribution within member cross sections, which can be

related to section shape factor (plastic-to-elastic modulus ratio, approximately 1.15

for wide flange sections) is also to be considered. For over-strength contribution

quantified by u/y, appointed as system over-strength, redistribution of internal

forces among the structural system members is the primary source.

By comparing the push-over curve response and the values provided in Table 11,

it may be concluded that global and system over-strengths, namely u and u/y,

for spatial frames are higher than for their perimeter counterparts.

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Table 11. Results from push-over curves (Triangular Distribution).

Perimeter Spatial

Frame du/dy Vy/VUBC Vu/Vy Vu/VUBC ULS du/dy Vy/VUBC Vu/Vy Vu/VUBC ULS

LA-3 4.71 2.63 1.35 3.55 Mech. (g) 4.48 3.00 1.97 5.27 Mech. (g)SE-3 2.64 3.24 1.24 4.03 Mech. (g) 2.86 3.91 1.49 5.81 3% (b)BO-3 2.76 3.79 1.26 4.77 3% (c) 3.21 3.39 1.64 5.60 3% (b)

LA-9 4.10 2.52 1.40 3.54 3% (b) 3.77 1.81 2.12 3.84 3% (b)SE-9 3.53 2.65 1.18 3.13 3% (b) 2.55 3.41 1.27 4.37 3% (b)BO-9 1.52 4.99 1.07 5.30 4% (1) 2.73 5.20 1.47 7.59 3% (b)

Keys: 3% = TPR; 4% = IDR; (b) = beam; (c) = column; g = global; (1) = first storey.



It is also interesting to note that, although the two alternative configurations

exhibit nearly similar yield strength, i.e. similar design over-strength y, perime-

ter frame curves possess a sharp transition between elastic branch and the sub-

sequent yield plateau. By contrast, such a transition is quite smooth for spatial

configurations. As was observed in recent studies [Gupta and Krawinkler, 2000a],

such a sharp transition is a consequence of the low relative importance of gravity

moments in the beams of perimeter frames (high ratio of seismic to gravity loads). It

is worth mentioning that the smoother the transition, the higher is the probability

of high local ductility demands.

The deformation parameters identified on pushover curves are roof drift val-

ues at the onset of yield (dy) and at ULS (du), respectively. Both perimeter and

spatial frames are characterised by a similar translation ductility (du/dy), with val-

ues ranging between 2.64 (SE-3P) and 4.71 (LA-3P) for low-rise frames, and 1.52

(BO-9P) and 4.10 (LA-9P) for medium rise frames. Furthermore, the storey drifts

experienced by nine-storey perimeter and spatial frames at similar values of dtopcan be explained on the basis of their different deformed shapes estimated at dyand du, respectively.

For spatial frames, the fulfilment of SCWB requirement was a specific design

target which led to a limitation of negative stiffness branch in the push over curves

(reduced P sensitivity). Indeed, the comparison between perimeter and spatial

BO frames, show that the latter reaches higher ultimate displacements than the

former, without showing significant strength loss. This result relates to low- and

medium-rise frames; nevertheless, it is critical in medium-rise frames (Fig. 4).

A further comparison of inelastic static performance for perimeter and spatial

frames may be derived by considering the governing ULSs, as listed respectively in

the sixth and eleventh columns of Table 11. Such ULSs are the onset of 3% plastic

rotations in beams and global mechanisms for perimeter and spatial typologies;

BO-3P exhibits 3% rotation in the columns, whilst BO-9P is characterised by a

high IDR.

It is instructive to correlate the values of the elastic critical multiplier for vertical

loads to the slope of the descending branch of push-over curves (Figs. 3 and 4).

As already observed, low-rise perimeter and spatial structures (three-storey) do

not suffer any instability problem since the critical multiplier of vertical loads is

consistently greater than 10. By contrast, some differences arise for the medium rise

frames (nine-storey). LA-9P and LA-9S possess nearly the same critical multiplier

(15.5 versus 16.5, respectively), thus negligible P effects are expected in both

cases. For BO-9S a significant improvement in the P resistance with respect to

the perimeter frame can be observed. Indeed, a comparison between the slope of the

softening branch of the perimeter and spatial curves shows a significantly reduced

slope for the BO-9S, corresponding to the increase in the critical multiplier value

from about 6 (perimeter) to about 8 (spatial). This improved structural response

is also confirmed by inelastic dynamic analyses, as discussed in the next section.

Although perimeter and spatial configurations behave in a similar fashion in terms



of global behaviour, e.g. same top displacement, the inter-storey drifts and plastic

rotation demands in beams and columns are lower for the spatial than for the

perimeter solution.

7. Inelastic Time History Analysis

The present study was focused on the non-linear global response and behaviour of

the structure rather than on the detailed performance of any of its elements. To

this end, emphasis was placed on investigating the beam and column maximum ro-

tations, storey displacements and storey drift ratios. Inelastic time history analyses

were performed on the set of three-storey and nine-storey perimeter and spatial

frames by means of the computer program DRAIN 2DX [Prakash et al., 1993]. The

aim is to evaluate and compare the seismic demand along the height for the sample

structures. The analyses were carried out by employing three acceleration histories

with different characteristics. The latter were derived from actual records of the El

Centro, Northridge and Kobe earthquakes.

It is worth mentioning that the design seismic loads adopted for the buildings

at three locations (LA, SE, BO) were quite different, since they correspond to three

different seismic hazard zones (namely, Z = 0.40 for LA, Z = 0.30 for SE, and

Z = 0.15 for BO). Nevertheless, the same ground motions, representative of severe

seismic input, were used for the seismic performance evaluation of the three sets of

buildings. The choice of this approach in the analyses is related to several reasons,

as specified in the following.

The choice of an accelerogram is a very complex task due to the uncertainties

involved. At the same site, as a result of the same source, the ground motions may

be very different in character for different events. The use of artificial ground mo-

tions, which can be generated with peak ground acceleration (PGA) matching those

specified for the seismic zone, may be questionable due to the poor frequency con-

tent of the generated acceleration history when compared to actual seismic records

[Broderick and Elnashai, 1996]. Furthermore, the use of scaled natural records,

obtained by either amplifying or reducing the PGA without changing other char-

acteristics (periods, duration, velocities, etc.) is also debatable [Bommer and Scott,

2000].

Finally, the analysis of the MRF design process, carried out in Secs. 4 and

5, has shown that stiffness, SCWB and stability, rather than strength (i.e. value

of the design base shear), are the governing design issues. Thus, a satisfactory

performance even under severe earthquakes, can be expected for MRF buildings

when the above design issues have been addressed, independently of the seismic

zone. Due to the fundamental period of vibration, i.e. 0.99 sec to 3.56 sec, a set

of accelerograms with near- and far-field effects were selected for the nonlinear

dynamic analyses. The latter were, however, conducted by using the same un-scaled

historical records for all MRF buildings.



Table 12. Characteristics of ground motions.

Earthquake Date Station Component Duration (sec) PGA (g)

Imperial Valley 18.5.40 El Centro S00E 53.80 0.348

Northridge 17.1.94 Newhall 340N/118W 59.98 0.583

Hyogoken-Nanbu 17.1.95 Kobe EW3 56.40 0.834

7.1. Ground motions

The main properties of the ground motions adopted in the time-history analyses

are summarised in Table 12. The selected ground motions cover a wide earthquake

scenario because of their different characteristics in terms of frequency content

and peak ground accelerations. The classical El Centro record exhibits a broad

frequency range, whilst the Northridge and Kobe ground motions are representative

of near-fault effects, especially in their few large pulses with a short duration.

By superimposing the 5% damping elastic spectra upon the acceleration spectra

of the considered records (Fig. 5) it may be observed that the UBC-94 spectrum

(soil type S2, relative to LA) under-estimates the ordinates derived from Northridge

and Kobe between 0.25 and 0.40 sec. By contrast, the smoothed spectra is more

conservative in the long period range (specifically for T > 2.0 sec).

7.2. Analysis results

7.2.1. Global drifts

Comparisons between maximum top displacements (adimensionalised to the total

building height, H) of perimeter and spatial frames computed via time history

analyses are provided in Fig. 6. Low-storey frames exhibit drift ratios greater than

1%; in fact, the values range between 1.16% and 3.75%. Perimeter MRFs are less

flexible than their spatial counterparts; the latter are characterised by an average

Dtot/H value equal to 2.44% versus 1.83% computed for the former. The drift ratios

vary as a function of the earthquake, El Centro and Kobe being respectively the

least and the most demanding. The values of Dtot/H for medium-rise perimeter

configurations range between 0.95% (LA-9P), and 1.55% (SE-9P). These values are

much smaller than the three-storey counterparts due to the longer periods of the

nine-storey frames and the shape of the spectral acceleration curves of the given

records.

It is interesting to point out that, except for LA, the maximum nondimensional

top displacements exceeds 1.00%. The highest values for the BO and SE frames are

induced by Newhall, whilst by Kobe for the LA frame. The drifts tend to be more

uniform and hence less dependent upon the input earthquake than the three-storey

frames. The range of variability is relatively narrow (1% to 1.5%).

The spatial configuration is characterised by roof displacements similar to

perimeter frames; by comparing the average values the difference is around 5%. The

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minimum and maximum values are 0.84% (LA-9S) and 1.52% (SE-9S), which are

comparable with the narrow range, e.g. 1% to 1.5%, evaluated earlier for perimeter

frames.

7.2.2. Inter-storey drifts

A comparison of inter-storey displacements is provided in bar-chart format in Fig. 7.

Inter-storey drift is adimensionalised with regard to inter-storey height (d/h).

The results show that the maximum drifts are generally lower than the thresh-

old value of 4%, corresponding to an inter-storey drift capacity (IDC) for MRFs

classified as special [FEMA 350, 2000]. Values beyond 4% occur for the Kobe earth-

quake, which frequently gives rise to high values of inelastic deformation demand.

The major issues mentioned earlier for top displacements are confirmed from the

drift results. However, a critical assessment of inter-storey drift values can pro-

vide further information on the localisation of the plastic demand throughout the

structural system.

Medium-rise spatial frames exhibit d/h values (Fig. 7) on average lower than

the perimeter ones. The results for the LA frame do not follow the same trend. The

inter-storey drifts computed for spatial configurations are always lower than for the

perimeter (except for LA). The scatter is lower for the spatial frames (1.40% and

3.50% are the minimum and maximum values of d/h) than for perimeter frames

(1.36% and 4.66% are the correspondent minimum and maximum d/h).

It is remarkable to highlight that in high (LA) and medium (SE) seismic areas,

the demand tends to be uniformly distributed, whilst a large variation characterises

the response of BO frames (low seismicity). The latter is due to the assumptions

adopted in the design. In the case of BO perimeter frames the SCWB require-

ments are not fulfilled, but they are fulfilled for the spatial configuration. Thus,

different plastic rotation demands are expected for such alternative configurations.

The envelope of the distribution of the plastic rotations along the height of BO-9P

shows that a concentration of plastic demand affects the first floor, thus leading to

a soft storey mechanism. Similar envelopes, for the other frames, demonstrate that

no soft mechanisms occur and the maximum plastic excursions affect the upper

storeys, without any detrimental effect on the global hysteretic dissipation.

The maximum inter-storey drifts for low-rise buildings are provided in Fig. 7.

By comparing the results, it may be observed that spatial frames exhibit higher

values of inter-storey drift (minimum d/h = 1.62% and maximum d/h = 5.06%);

the highest values, greater than 4.5%, occurs for the Kobe record. For the perimeter

frames a reduced variation of drift values is found. LA-3P experiences d/h = 1.70%

for the El Centro record, whilst the maximum value is 3.83% and is due to BO-3P

(Kobe ground motion).

7.2.3. Plastic rotations

Maximum plastic rotations for both perimeter and spatial typologies and low-to-

medium rise buildings are provided below with regard to the beams (Fig. 8) and

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Fig. 7. Inter-storey drifts for perimeter and spatial MRFs: three-storey (left) and nine-storey (right).

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columns (Fig. 9). In all bar charts the threshold of 0.03 rad (3%), target plastic

rotation (TPR) of beam-to-column connections as defined within the SAC Steel

Project [FEMA 350, 2000], is drawn.

The end beam rotations for medium rise perimeter frames (Fig. 8) range from

a lower bound of 1.01% (LA-El Centro) to an upper limit of 3.33% (BO-Kobe).

Kobe is the most severe earthquake case: for all seismic zones the beams undergo

plastic rotations greater than 2%. By contrast, the least demanding record is El

Centro; it gives rise to plastic rotations of 1.07% on BO, 1.01% for LA and 1.12%

in SE frame. Intermediate values ranging between 1.25% (LA) and 2.22% (SE) are

induced by Newhall ground motion.

A smoothed variation characterises the plastic demands on nine-storey spatial

frames. The minimum plastic rotation was computed for BO-El Centro (1.23%)

while the maximum (3.15%) is reached in the LA frame when the Kobe record is

considered. The two sets of values show that, except for Kobe, the plastic rotation

demands are lower than the 3% TPR required by SAC. The latter is a direct

consequence of the SCWB fulfilment, adopted as a specific design target for spatial

frames.

Maximum plastic rotations in low-rise perimeter and spatial MRFs are also

provided (Fig. 8). The values of inelastic demands in the beams of perimeter and

spatial MRFs are quite close and do not generally overcome the 3% limit value.

The Kobe earthquake provokes the largest plasticity in the beams of BO and SE

frames (respectively 5.13% and 5.50%). These results may be justified in the light

of the frequency content of Kobe. It is particularly severe in the neighbourhood of

natural periods of low-rise frames. High values of beam rotations can be found in

the case of perimeter as well as spatial frames; they were deliberately designed with

similar fundamental periods.

For the sake of completeness, the maximum plastic rotations in the column

elements of the nine- and three-storey frames, both perimeter and spatial, are also

considered. In the case of perimeter configurations (Fig. 9) the nine-storey Boston

frame is characterised by relevant plastic demands, especially with regard to the case

of the Newhall record (maximum rotation 3.53%), that corresponds to the highest

demand for the perimeter configuration. The behaviour of the column elements

of the LA perimeter frame is almost elastic, since the maximum value is equal to

0.08%. The comparison with the spatial configurations suggests that for all but the

LA and SE frames under the Kobe ground motion, the columns are in the elastic

range.

The values for the column plastic rotations of the three-storey frames are also

shown in Fig. 9. The scenario outlined is quite different from the aforementioned,

with regard to the spatial frames where the adopted design philosophy is slightly

different from that adopted for the nine-storey spatial frames. While the plastic

rotations required in the columns of the spatial configuration are lower than in

perimeter layouts, some plastifications in the columns occur. Nevertheless, since

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x

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0.05

0.06


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Newhall

Kobe

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m

a

x

(

r

a

d

)

Fig. 9. Column plastic rotations perimeter and spatial MRFs: three-storey (left) and nine-storey (right).



the design was based upon the fulfilment of SCWB, the maximum plastic rotations

take place in the base column ends.

The comparison between inter-storey drifts (Fig. 7) and beam plastic rotations

(Fig. 8) shows close values for both set of data (low-to-medium rise and perimeter-

spatial), especially for SCWB frames. It is, therefore, confirmed, as in previous

studies [Mele et al., 1995], that plastic rotation can be predicted via the inter-

storey drift index. Inelastic inter-storey drift can be related to the elastic inter-

storey drift by means of the displacement amplification factor Cd, thus a reasonable

prediction of plastic rotation demands can be obtained by means of a simple elastic

analysis.

7.2.4. Hysteretic energy

Inelastic time history analyses also indicate the hysteretic energy dissipated during

the ground motions. In Fig. 10 the energy time histories for perimeter and spatial

Fig. 10. Hysteretic energy diagrams: three-storey (left) and nine-storey (right) MRF buildings.



MRF buildings, under the El Centro, Newhall and Kobe seismic inputs, are pro-

vided. They were computed by considering the energy dissipated by each MRFs and

accounting for the actual number of frames used to withstand horizontal forces.

It can be observed that the energy dissipated by the three pairs (LA, SE and BO)

of buildings configurations (perimeter, spatial) is quite close. The total dissipated

energy is a function of the building natural period and of the earthquake ground

motion.

With regard to the ground motion, Newhall represents the worst case scenario

for the set of assessed buildings: the energy varies in the range 5000 kNm (BO-

3) and 10 000 kNm (LA-9). Due to the higher values of beam plastic rotations

(greater than 4%) and inter-storey drifts (around 3%) SE-3 MRFs dissipate a large

amount of energy; the maximum values are comparable to the LA counterparts.

These results were also found for medium-rise buildings; in this case, though, the

influence of higher-mode effects gives rise to more complex responses. Concerning

the dependence of the hysteretic energy on the building natural period, it may be

observed that lower energy values can be found for longer period buildings (e.g.:

BO-9 versus LA-9), as a consequence of a sort of self-isolation effect. As a result,

the energy dissipated by the three pairs (LA, SE and BO) of buildings configurations

(perimeter, spatial) is quite close. Therefore, it can be argued also that in terms of

hysteretic energy, the dynamic response of the perimeter and spatial MRFs is fairly

similar; thus, their equivalent behaviour is confirmed.

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