Bulletin of Earthquake Engineering 2: 221–259, 2004.
© 2004 Kluwer Academic Publishers. Printed in the Netherlands.
Seismic Performance of RC Frames Designed
to Eurocode 8 or to the Greek Codes 2000
TELEMACHOS B. PANAGIOTAKOS and
MICHAEL N. FARDIS∗
Structures Laboratory, Department of Civil Engineering, University of Patras, P.O. BOX
1424, GR26500, Patras, Greece
∗
Corresponding author. Tel: +302610 997651, Fax: +302610 997694, E-mail:
fardis@upatras.gr.
Abstract. For the first time after the finalisation of the European Norm for seismic design
of buildings (Eurocode 8 – EC8), the performance of RC buildings designed with this code is
evaluated through systematic nonlinear analyses. Regular 4-, 8- or 12-storey RC frames are
designed for a PGA of 0.2 or 0.4 g and to one of the three alternative ductility classes in
EC8. As the Eurocodes are meant to replace soon existing national codes, design and performance is also compared to that of similar frames designed with the 2000 Greek national
codes. The performance of alternative designs under the life-safety (475 years) and the damage limitation (95 years) earthquakes is evaluated through nonlinear seismic response analyses. The large difference in material quantities and detailing of the alternative designs does
not translate into large differences in performance. Design for either Ductility Class High
(H) or Medium (M) of EC8 is much more cost-effective than design for Ductility Class Low
(L), even in moderate seismicity. It is also much more cost-effective than design to the 2000
Greek national codes.
Key words: collapse prevention, damage limitation, drift control, ductility, earthquakeresistant design, Eurocode 8, life safety, RC buildings, RC frames, seismic design
1. Introduction: Design for Ductility According to Eurocode 8
Between 1992 and 1998 about 60 Structural Eurocodes were published by
CEN as European prestandards (ENV). 1998 has seen the start of the
work for their revision and conversion to the first generation of European
standards (ENs). It is expected that by 2005 all Structural Eurocodes will
be approved and made available by CEN in its three official languages.
Then the EN Eurocodes will be the recommended European codes for the
structural design of civil engineering works and of their parts. Nonetheless,
and at least for this first generation of EN Eurocodes, the use of national
regulations will also be allowed, as long as EU Member States maintain
them for parallel use with the EN Eurocodes. Unlike national regulations,
national standards that are not fully compatible with the EN Eurocodes
will have to be withdrawn before the end of the present decade.
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TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
The EN Eurocodes have certain built-in flexibility on key items controlling the level of safety, serviceability and durability offered by structures
designed according to them, taking into account economy. These aspects
are considered to lie within national authority. To facilitate national choice
on these aspects without sacrificing harmonisation of structural design
codes at the European level, as well as to accommodate geographic, climatic, etc., differences (including differences in the seismotectonic environment), the system of “Nationally Determined Parameters” (NDPs) has
been devised and adopted in the EN Eurocodes. NDPs include symbols
(e.g., safety factors, the mean return period of the design seismic action,
etc.), technical classes (e.g., ductility classes), or even procedures or methods (e.g., alternative models of calculation). Alternative classes or procedures/methods considered as NDPs are fully described in the normative
text of the EN Eurocode. For NDP-symbols, the EN Eurocode may give
an acceptable range of values and will normally recommend in a nonnormative note a value to be used. It may also recommend a class or a
procedure/method, among the alternatives identified in the EN Eurocode
text as NDPs.
National choice regarding the NDPs will be exercised through the
National Annexes, to be developed and published by each EU Member
State – as integral parts of the national versions of the EN Eurocodes –
within 2 years from the publication by CEN of the approved EN Eurocode.
National Annexes may also contain country-specific data (e.g. seismic zonation maps, spectral shapes for the various types of soil profiles provided
in EC8, etc.), which will constitute also NDPs. In these two years, Member States will be expected to make the national choice for the NDPs, so
that design with the EN Eurocodes will provide the national target level of
safety at a level of economy which is nationally acceptable (implying that
design with the EN Eurocodes will not be unduly less cost-effective than
with the existing national standards or regulations). Member States will be
advised by the European Commission to adopt for the NDPs the choices
recommended in the notes of the EN Eurocode, so that the maximum feasible harmonisation across the EU is achieved. If the National Annex does
not exercise national choice for some NDPs, such choice will be the responsibility of the designer, taking into account the conditions of the project
and other national provisions.
EN1998-1 (Eurocode 8: “Design of Structures for Earthquake Resistance”, in short: EC8) has been technically and editorially finalized and
was submitted to the European Committee for Standardisation (CEN) in
November 2003 for approval by the CEN national member bodies (CEN,
2003).
EC8 provides (as NDP) for three classes of ductility (DC) of RC buildings. In the lower ductility class (Low or L), seismic design does not rely
SEISMIC PERFORMANCE OF RC FRAMES
223
on energy dissipation but on elastic response; overstrength is accounted for
by reducing elastic force demands by a behaviour factor q (equivalent to
the force reduction factor R of US codes) equal to 1.5. Dimensioning and
detailing is just to the Eurocode for concrete structures without earthquake
resistance (Eurocode 2, in short: EC2), with the earthquake considered as
lateral loading (as, e.g., the wind). EC8 recommends design with ductility
class L only for low seismicity regions (suggested limit of the peak ground
acceleration for the use of DC L: 0.1 g).
In the two upper DCs of EC8 (Medium or M, and High or H) design
is based on energy dissipation and on ductility and aims at controlling the
inelastic seismic response via:
– The structural configuration and the relative sizing of members (use of
shear walls, strong column/weak beam frames, etc).
– Detailing of plastic hinge regions to safely accommodate the corresponding inelastic deformation demands. This is achieved by relating quantitatively deformation demands (e.g., the curvature ductility factor) in these
regions to the behaviour factor q that reduces the elastic spectrum for
design based on linear elastic analysis (the R of US codes).
Design of multistorey RC frames to DC M or H for energy dissipation
comprises the following:
• Fulfillment of the strong column/weak beam rule, with an overstrength
factor of 1.3 on beam flexural capacities.
• Member verification in terms of forces and resistances for the Ultimate
Limit State (ULS) under the design earthquake (the return period of which
is an NDP with a recommended value of 475 years), with the elastic spectrum reduced by a behaviour factor q equal to 3 times an overstrength
factor αR – for frame redundancy – in DC M, or to 4.5 times the overstrength factor αR in DC H. Unless it is estimated via pushover analysis,
the overstrength factor αR may be taken equal to a default value of 1.3 for
planwise-regular multistorey multibay frames, or to the average of: (a) 1.0
and (b) that default value for planwise-irregular ones. Heightwise-irregular
frames are penalized with a 20% reduction of the value of the behaviour
factor.
• Damage limitation under an occasional earthquake, which is an NDP
with a recommended value of 95 years for its return period, or with a
recommended scale factor of 0.5 on the elastic spectrum of the design
(475 years) earthquake. Damage limitation consists in controlling the storey
drift ratio up to a limit of 0.5% for brittle nonstructural infills in contact
with the RC frame, assuming 50% of uncracked gross section rigidities.
• Capacity design of beams and columns against pre-emptive shear failure.
Design shears are derived from the flexural capacities of the plastic hinges
forming around the joints at the member ends (e.g., if plastic hinges form
224
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
in beams, the capacity design shear of the column is based on the beam
flexural capacities) and include an overstrength factor from 1.0 (for beams
of DC M) to 1.3 (for columns of DC H).
• Member detailing, including meeting the curvature ductility factor
demands corresponding to the value of the behaviour factor q.
Dimensioning for the ULS against ductile, “deformation controlled”, failure modes for the seismic action effects from the analysis (2nd bullet point
above) applies to bending at beam ends, at the base section of columns and
at top-storey columns. At all other column locations, flexural capacity may
be conditioned by the strong column-weak beam rule.
Local deformation and ductility demands to be sustained by those members or parts thereof intended to develop inelastic action, increase as the
DC (and with it the value of q) increases. Deformation- and ductilitycapacity is a function of the ductility of the materials and of the configuration and detailing of the members. Tables I and II summarise the main
detailing and dimensioning rules in EC8, for beams, and columns of the
three ductility classes.
Detailing rules to control the deformation capacity of plastic hinges in
concrete members through the amount of compression reinforcement ρ
in beam end sections (see Table I) or of the confining reinforcement, ωwd ,
in columns (see Table II), are linked analytically to the local curvature ductility factor, µφ , and through it to the value of q (prior to any reduction
due to irregularity in elevation):
µφ = 2q − 1
if T1 ≥ TC ,
µφ = 1 + 2(q − 1)TC /T1
(1)
if T1 < TC .
(2)
where T1 the fundamental period of the building and TC the transition
period between the constant acceleration region and the constant pseudovelocity regions of the spectrum. Eqs. 1 and 2 are based on a relation
between µφ and the displacement ductility factor, µδ : µφ = 2µδ − 1, which
is normally a conservative approximation for RC members, and on the
Vidic et al. (1994) relation between µδ and q:
µδ = q
if T1 ≥ TC,
µδ = 1 + (q − 1)TC /T1
if T1 < TC .
(3)
So the continuous spectrum of q-values resulting from the dependence of
q on the overstrength factor αR , produces detailing that also varies continuously.
Rules for confinement of column end regions do not apply uniformly to
all column ends, but only where plastic hinges are meant to develop. In end
regions of columns protected from plastic hinging through capacity design
6dbL , hb /4, 24dbw , 0.175m
6 mm
8dbL , hb /4, 24dbw , 0.225 m
√
0.08 fck (MP a)/fyk (MP a)(0)
0.75d
–
–
–
Transverse bars (w):outside critical regions
Spacing sw ≤
0.8d, 0.3 m if: VEd < VRd2 /5
0.6d, 0.3 m if:
VRd2 /5 < VEd < 2VRd2 /3
0.3d, 0.2 m if: 2VRd2 /3 < VEd
ρw ≥
0.0011 (S500, C25–C35)
In critical regions
dbw ≥
8 mm
Spacing sw ≤
10dbL , hb /3, 20dbw , 200 mm
(b) exterior joint
–
–
yd
≤ 6.25(1 + 0.8νd ) ffctm
yd
(1+0.75 ρmax )
d ) fctm
≤ 6.25(1+0.8ν
ρ
f
0.26fctm /fyk ,
0.13%(0)
0.04
–
–
–
hw
EC8 DCL
–
0.5fctm /fyk
EC8 DCM
ρ +0.0018fcd /(µφ εsy,d fyd )(1)
214 (308 mm2 )
As,top−supports /4
0.5As,top
1.5hw
EC8 DCH
0.65(fcd /fyd )(ρ /ρ)+0.0015<(7/fyd )
212 (226 mm2 )
As,top−supports /4
0.5As,top
0.5fctm /fyd
Longitudinal bars (L)
ρmin , tension side
ρmax , critical regions(1)
As,min, top & bottom
As,min ,top-span
As,min , cr. regions bottom(2)
dbL /hc -bar crossing joint(3)
(a) interior joint
2hw
“Critical region” length
Greek code
Table I. Rules in EC8 and in the Greek codes for detailing and dimensioning of beams
SEISMIC PERFORMANCE OF RC FRAMES
225
Rb
1.2 M
± Vo,g+ψ2 q
lc
VRd,s = bw zρw fywd + Vcd
with Vcd = 0.3VRd1
Greek code
1.2
MRb
(4)
± Vo,g+ψ
lc
2q
EC8 DCH
MRb
lc
EC8 DCM
EC8 DCL
(0) NDP (Nationally Determined Parameter) according to EC2. Table gives the recommended value.
(1) µφ is the curvature ductility factor corresponding to the value of the behaviour factor, q (see Eqs. (1), (2)).
(2) Additional to compression steel needed for the ULS verification for MEd .
(3) hc is the column depth in direction of bar and νd is the column axial load
ratio. (4) At a member end where
the
moment
capacities
around
the
joint
satisfy:
MRb > MRc , MRb is replaced in the calculation of the design
shear force, VEd , by MRb ( MRc / MRb )
(5) VEmax , VE,min are the algebraically maximum and minimum values of VEd resulting from the ± sign; VEmax is the absolutely largest of the
two values, and is taken positive in the calculation of ζ ; the sign of VEmin is determined according to whether it is the same as that of VEmax
or not.
VEd , seismic(4)
VRd,s , critical regions
(4)
± Vo,g+ψ
From the analysis
2q
VRd,s = Vwd = bw zρw fywd
As in EC2:
–
Vcd = 0
VRd,s = Vwd = bw zρw fywd cot θ
1 ≤ cot θ ≤ 2.5, Vcd = 0
Inclined bars at angle ±α If 2.5 > VEmax /(2 + ζ )fctd bw d > 1.125: If VEmax /(2 + ζ )fctd bw d > 1:
–
to beam axis, with crossAs = 0.5VEmax /fyd sin α
As =VEmax /fyd sin α
section As /direction(5)
& stirrups for 0.5VEmax .
& stirrups for 0.5VEmax
If: VEmax /(2 + ζ )fctd bw d >:
As = 0.5VEmax /fvd sin α
Shear design
Table I. Continued.
226
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
0.25 m
sw ≤(3),(4)
ωwd ≥(5)
αωwd ≥(6),(7)
8dbL , 0.5 min(hc , bc ), 0.1m
0.10
0.85νd (0.35(Ac /Ao )+0.15) − 0.035
“critical region” length
max(hc , bc ), 0.6m, lc /5
Longitudinal bars (L):
ρmin
0.01
ρmax
0.04
dbh ≥
14 mm
Bars per side ≥
2
spacing of restrained bars
< 0.2 m
distance of unrestrained
bar from restrained
Transverse bars (w)outside critical regions
dbw ≥
6 mm, dbL /4
spacing sw ≤
12dbL , min(hc , bc ), 0.3 m
sw in splices ≤
6dbL
In critical regions:(2)
dbw ≥(3)
8 mm, dbL fyd /fywd /3
X-section sides, hc , bc ≥
Greek code
6dbL , bo /3, 0.125 m
0.08
30µ∗φ νd εsy,d bc /bo − 0.035
0.01
0.04
8 mm
3
≤0.15m
≤0.15m
0.25 m;
hv /10 if θ=Pδ/V h>0.1(1)
1.5max(hc ,bc ), 0.6 m, lc /5
EC8 DCH
Table II. Rules in EC8 and in the Greek codes for detailing and dimensioning of columns
–
2
≤0.2 m
8dbL , bo /2, 0.175 m
0.4dbL fyd /fywd
–
–
–
–
0.1Nd /Ac fyd , 0.2%(0)
0.04(0)
6 mm, dbL /4
20dbL , min(hc , bc ), 0.4 m
12dbL , 0.6 min(hc , bc ), 0.24 m
–
–
EC8 DCL
max(hc ,bc ), 0.6 m, lc /5
EC8 DCM
SEISMIC PERFORMANCE OF RC FRAMES
227
VEd seismic(10)
Crit. region at column base
ωwd ≥
αωwd ≥(8)
Capacity design check at
beam-column joints:
Verification for Mx -My -N:
Axial load ratio νd = NEd /
Ac fcd
Shear design
Table II. Continued.
1.4
ends
MRc
lc
1.3
ends
MRc
lc
(9)
1.1
ends
MRc
lc
(9)
from the analysis
Truly biaxial or uniaxial with γ ια (Mz /0.7, N), (My /0.7, N )
≤ 0.55
≤ 0.65
–
–
–
–
EC8 DCL
Biaxial
<0.65
EC8 DCM
0.12
0.08
30µφ νd εsy,d bc /bo − 0.035(8)
(9)
1.3 MRb ≤
MRc
EC8 DCH
0.10
0.85νd (0.35(Ac /Ao )+0.15)−0.035
1.4 MRb ≤
MRc
Greek code
228
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
VRd,s = bw zρw fywd + Vcd
If νd < 0.1 : Vcd = 0.3VRd1
If νd > 0.1 : Vcd = 0.9VRd1
Greek code
EC8 DCH
EC8 DCL
As in EC2: VRd,s = Vwd = bw zρw fywd cot θ
1 ≤ cot θ ≤ 2.5, Vcd = 0
EC8 DCM
(0) Note (0) of Table V applies.
(1) hv is the distance of the inflection point to the column end further away, for bending within a plane parallel to the side of interest; lc is
the column clear length.
(2) For DCM: If a value of q not greater than 2 is used for the design, the transverse reinforcement in critical regions of columns with axial
load ratio νd not greater than 0.2 may just follow the rules applying to DCL columns.
(3) For DCH: In the two lower storeys of the building, the requirements on dbw , sw apply over a distance from the end section not less than
1.5 times the critical region length.
(4) Index c denotes the full concrete section and index o the confined core to the centreline of the hoops; bo is the smaller side of this core.
(5) ωwd is the ratio of the volume of confining hoops to that of the confined core to the centreline of the hoops, times fyd /fcd .
(6) α is the “confinement effectiveness” factor, computed as α = αs αn ; where: αs = (1−s/2bo )(1−s/2ho ) for hoops and αs = (1−s/2bo ) for spirals;
αn = 1 for circular hoops and αn =1−bo /[(nh −1)ho ]+ho /[(nb −1)bo ]/3 for rectangular hoops with nb legs parallel to the side of the core with length
bo and nh legs parallel to the one with length ho (cf. Eq. 6).
(7) For DCH: at column ends protected from plastic hinging through the capacity design check at beam-column joints, µ∗φ is the value of the
curvature ductility factor that corresponds to 2/3 of the basic value, qo , of the behaviour factor used in the design (see Eqs. 1, 2); at the ends
of columns where plastic hinging is not prevented because of the exemptions listed in Note (10) below, µ∗φ is taken equal to µφ defined in Note
(1) of Table I (see also Note (9) below); εsy,d = fyd /Es .
(8) Note (1) of Table I applies.
(9) The capacity design check does not need to be fulfilled at beam-column joints: (a) of the top floor, (b) of the ground storey in two-storey
buildings with axial load ratio νd not greater than 0.3 in all columns, (c) if shear walls resist at least 50% of the base shear parallel to the
plane of the frame (wall buildings or wall-equivalent dual buildings), and (d) in one-out-of-four columns of plane frames with columns of
similar size.
(10) At a member end where the moment capacities around the joint satisfy:
MRb < MRc , MRc is replaced by MRc ( MRb / MRc ).
VRd,s seismic
Table II. Continued.
SEISMIC PERFORMANCE OF RC FRAMES
229
230
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
in flexure at beam/column joints, only the prescriptive rules – e.g., against
buckling of rebars, etc. – do apply. For DC H frame buildings in particular, confinement requirements in such regions are reduced to two-thirds of
those applying where plastic hinges are expected.
It should be noted that the q-factors allowed in EC8 for DC H, M or
L RC frames are much lower than the R-factors of US codes for Special,
Intermediate or Ordinary RC frames, respectively, although detailing and
capacity design requirements for DC H, M or L frames are, in general,
more demanding than for the US Special, Intermediate or Ordinary frames,
respectively.
In this paper regular RC frames designed according to the three alternative Ductility Classes of EC8 for a design earthquake with peak ground
acceleration (PGA) of 0.2 or 0.4 g, are evaluated via nonlinear timehistory analyses at the collapse-prevention and at the damage-limitation
performance levels, under the design earthquake and under the damage
limitation earthquake (taken with a scale factor of 0.5 on the elastic spectrum of the design earthquake, as allowed by EC8), respectively. A design
PGA as high as 0.4 g is certainly not meant to represent the seismicity of
many regions in Europe; nonetheless, with such a high PGA seismic design
will control and its effects on the response will be more clear.
To compare the cost-effectiveness of EC8 to that of one of the national
codes it is intended to replace, the design and performance of the EC8
frames are compared to those of similar frames designed to the 2000 Greek
concrete and seismic design codes, EKOS 2000 and EAK 2000, respectively.
The set of these two codes is considered to be at the forefront of national
codification for earthquake resistance of concrete buildings, as it represents a development from and an improvement over the 1994 pre-standard
(ENV) versions of EC2 and EC8. Moreover, as their earlier and slightly
different versions have been in mandatory use since 1995, these codes have
been proven in practice as complete and operational, and – to the extent
they had been applied in the region hit by the 1999 Athens earthquake –
as safe too.
A major difference of the Greek codes from the EN Eurocode 8 is that
in the falling branch of the spectrum design spectral accelerations fall as
T −2/3 , instead of 1/T . Although the drift limit for damage limitation is the
same, the damage limitation earthquake is taken as 40% of the design seismic action, instead of the 50% value recommended in EC8; the slightly
greater stiffness employed by the Greek codes increases the effect of this
difference on the design. Two Ductility Classes are provided for concrete
buildings. The first one essentially coincides with DC L in EC8. In the
other, which is based on energy dissipation, the q factor is equal to 3.5
regardless of irregularity. As shown in Tables I and II; capacity design
rules are more demanding than in EC8, even for DC H; rules for member
SEISMIC PERFORMANCE OF RC FRAMES
231
minimum and maximum reinforcement are typically less demanding than
their EC8 counterparts, except for column confinement, where they are
normally more demanding than in EC8, DC H.
2. Design of Multistorey RC Frames to EC 8 (DC L, M, H) or to the
Greek Code
RC frames with 4, 8, or 12 storeys of 3 m height were designed to EC2
and EC8, for the Type 1 spectrum recommended in EC8 for soil type C
(stiff soil, with transition period between the acceleration- and velocitycontrolled regions of 0.6s) and a PGA of either 0.2 or 0.4 g. For this type
of soil the PGA at grade level is obtained by multiplying the PGA on rock
with an S factor of 1.15; therefore the frames are designed to a PGA on
rock of 0.2/1.15 = 0.175 g, or 0.4/1.15 = 0.35 g, respectively. Three frames
with the same number of storeys are designed for each PGA value to EC8,
one for each of the three ductility classes. A fourth one is designed according to the Greek codes, for the same elastic response spectrum (i.e., for the
soil C spectrum and a PGA of either 0.2 or 0.4 g).
Concrete of class C30/37 (nominal cylindrical strength of 30 MPa) and
class B S500 steel (relatively ductile Tempcore type of steel, with nominal
yield stress of 500 MPa) are assumed.
The frames have three 5 m-long bays. A two-way system of beams with
a span of 5 m is considered in both horizontal directions. The slab is 0.15 m
thick and in the design is considered to contribute to the moment of inertia
of the beams with an effective flange width according to EC2.
In addition to the self weight of the beams and the slab, a distributed
dead load of 2 kN/m2 due to floor finishing and partitions and imposed
live load with nominal value of 1.5 kN/m2 are considered. In the combination of gravity loads (“persistent design situation”) nominal dead and live
loads are multiplied with load factors of 1.35 and 1.5, respectively. Following EC8, in the seismic design situation, dead and live loads enter with
their nominal value and with 30% of the nominal value, respectively.
Frame columns are square; their side length hc is the same in all storeys
but is smaller in the two exterior columns, so that their uncracked grosssection stiffness is about half that of the interior ones. In this way elastic
seismic chord rotation demands at the two beam ends in the exterior bays
of the frame are approximately equal (and, to the extent the equal displacement approximation holds at the level of member chord rotations, inelastic chord rotation demands are about equal at the two ends of exterior
beams). Beams have the same web width (bw = 0.3 m for the 0.2 g design
PGA, bw = 0.35 m for the 0.4 g PGA) in all storeys but different depth, hb .
232
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
Design is based on the results of linear elastic (equivalent) static
analysis, for lateral earthquake forces distributed over the height according to an assumed linear mode shape (termed “lateral force procedure” in
EC8). Because such a static analysis systematically overestimates the results
of a more representative modal response spectrum analysis, EC8 allows
multiplying its results by 0.85. This reduction factor is applied for the EC8
designs, but not for those according to the Greek codes. Lateral forces are
derived from the design spectrum (5%-damped elastic spectrum divided by
the behaviour factor q) at the fundamental period of the building, which
is estimated through the Rayleigh quotient on the basis of the storey elastic horizontal displacements resulting from equivalent lateral forces with
(inverted) triangular distribution. 50% of the uncracked gross section stiffness is considered in the EC8 designs, whereas slightly higher stiffnesses are
employed according to the Greek codes.
Design neglects any torsional effects due to accidental eccentricity and
simultaneous action of the two horizontal components of the earthquake
(according to the familiar 0.3:1 rule), as the nonlinear analyses of the
response of the frames to the design earthquake are performed in 2D,
under only one component of the seismic action and without accidental
eccentricity.
In the analysis the columns of the bottom storey are assumed fixed at
grade level. The finite size of beam-column joints is considered, but joints
are assumed rigid. P − effects are neglected. Beam gravity loads are computed on the basis of beam tributary areas in the two-way square slab system.
At a preliminary design stage, the (uniform) column depth hc and the
beam depths hbi at each storey are tailored to the interstorey drift ratio
limitation of 0.5% (for brittle nonstructural infills) for the damage limitation earthquake (taken as 50% of the design earthquake, i.e., with a PGA
on Type C soil of 0.10 or 0.20 g). Member sizing takes place iteratively, via
a simplified analysis in which inflection points due to the lateral earthquake
loading are assumed at beam mid-span and at column mid-height. After
assuming small initial member sizes, the (constant throughout the building) column depth hc is chosen to fulfil the 0.5% drift limit at the storey with the minimum interstorey drift among those violating this limit. In
other storeys, beam depth may need to be increased until the 0.5% drift
limitation in that storey is also fulfilled. The estimate of the fundamental
period by the Rayleigh quotient and the resulting elastic spectral values are
revised during the iterations. Member depths are rounded up to the nearest
50 mm. Internal forces to be used for the calculation of member reinforcement for the ULS in bending, are obtained from a linear elastic analysis
of the full model of the so-sized frame. During the dimensioning of beam
reinforcement, beam depths in some storeys may increase further, to respect
233
SEISMIC PERFORMANCE OF RC FRAMES
Table III. Column sizes and normalized axial load: νd = N/Ac fcd at base due to gravity
loads
Building
4-storey
8-storey
12-storey
Design
All designs
EC8 DC L
EC8 DC M or H;
Greek code
EC8 DC L
EC8 DC M or H;
Greek code
Design PGA : 0.20g
Design PGA : 0.40g
Exterior
columns
Interior
columns
Exterior
columns
Interior
columns
hc (m)
νd
hc (m)
νd
hc (m)
νd
hc (m)
νd
0.45
0.55
0.55
0.051
0.075
0.075
0.55
0.65
0.65
0.078
0.115
0.115
0.55
0.75
0.55
0.077
0.098
0.153
0.65
0.90
0.65
0.122
0.138
0.240
0.85
0.85
0.065
0.065
1.00
1.00
0.086
0.086
1.00
0.60
0.110
0.198
1.20
0.70
0.138
0.314
the maximum top steel ratio at beam supports to columns. In such cases
the linear elastic analysis is repeated, for conformity of the fundamental
period and of the analysis results with the final member depths.
In most cases member sizing for the “damage limitation earthquake”
controls over ULS dimensioning requirements under the design earthquake. It is possible that in EC8 designs member sizes would have further
increased had the “modal response spectrum analysis” been used, which
could have led to increases in some storey drifts, especially in the 12-storey
frames.
Final column depths (the same for DC M and DC H) are listed in
Table III, along with the resulting axial load ratios, νd = N/Ac fcd , at the
base of the column under the gravity loads which are considered to act
simultaneously with the design earthquake. Final beam depths, hb , top and
bottom beam reinforcement ratios, ρ and ρ (average at the two ends) and
column total steel ratio ρc for each story, are given in Figures 1–3.
The νd values in Table III are low with respect to the limits listed at the
4th row from the bottom of Table II. As these limits apply for the sum of
the axial loads due to the design earthquake and the gravity loads considered to act simultaneously with it, the νd values in Table III leave ample
margin for the axial load due to the design earthquake itself. It was pursued to have the same column sizes in all different designs of the same
frame for given PGA. In the 12-storey frames with design PGA of 0.2 g,
keeping column size the same for DC L as in the other designs gives oversized columns for these other designs and excessive beam reinforcement
ratios for EC8 DC L. In the 8- or 12-storey frames designed for a PGA of
234
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
5
5
(a) 5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
Storey
1
Exterior
Interior
Exterior
Interior
Exterior
Interior
DC-L
0
5
0
5
0
5
0
5
Exterior
Exterior
Interior
Exterior
Interior
Interior
4
4
4
3
3
3
3
2
2
2
2
1
1
1
Storey
4
1
Storey
DC-M
0
0
5
5
0
0
5
5
Exterior
Exterior
Interior
Interior
Exterior
Interior
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
DC-H
0
0
5
5
0
0
5
5
Storey
Exterior
Interior
Exterior
Interior
Exterior
Interior
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
EAK
0
0
0.0
0.4
0.2
0.6
0.0
0.5
1.0
1.5
2.0
0
0.0
0
0.5
1.0
1.5
0.0
0.5
1.0
1.5
2.0
hb (m)
Figure 1. Beam depth (hb ), beam top (ρ) and bottom (ρ ) reinforcement ratio and column total reinforcement ratio (ρc ) in 4-storey frames for EC8 DC L (1st from top), M
(2nd from top) or H (3rd from top) and for Greek code EAK (bottom). Design PGA:
(a) 0.2 g; (b) 0.4 g. Closed circles: exterior and open circles: interior members.
0.4 g, it was not feasible to keep the same column size for EC8 DC L as in
the other designs. With this constraint removed the common column sizes
of the 0.20 g 12-storey frames are smaller than in the 0.40 g DC L frames
but larger than those of the three other designs.
Table IV lists the value of the frame fundamental period, T1 , and of the
design base shear, Vb , of the four alternative frame designs for each design
PGA. Differences between the value of T1 of the design to the Greek codes
and those of the alternative designs to EC8 DC M or H reflect not so
much differences in member sizes, but the systematically greater stiffness
235
SEISMIC PERFORMANCE OF RC FRAMES
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
Storey
(b) 5
5
5
Exterior
Interior
1
Exterior
Interior
Exterior
Interior
DC-L
0
5
0
5
Storey
Exterior
Interior
0
5
Exterior
Interior
0
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
Exterior
r
Interior
DC-M
0
5
0
5
Storey
Exterior
Interior
0
5
Exterior
0
5
Interior
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
Exterior
Interior
4
Storey
DC-H
0
0
5
5
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0.0
0
2.5 0.0
1
0
0
Exterior
Interior
5
Exterior
Interior
4
5
Exterior
Interior
4
EAK
0
0.0
0.2
0.4
0.6
0.5
1.0
1.5
2.0
0.5
1.0
1.5
0
2.0 0.0
0.5
1.0
1.5
2.0
Figure 1. Continued.
assumed by the Greek codes as a fraction of that of the uncracked gross
concrete section. Differences between the values of the design base shear in
the four alternative frame designs reflect mainly differences in the value of
the behaviour factor, q. All EC8 designs incorporate in their design base
shear values the reduction factor of 0.85, due to the application in the
analysis of the (equivalent static) lateral force procedure. For frames with
fundamental period, T1 , above the value of the transition period, TC , the
higher value of the base shear of the Greek code designs compared to
that of their EC8 DC M counterparts reflects also the shorter fundamental
period of the Greek code designs, as well as the decay of the falling branch
of the spectrum as T −2/3 , instead of 1/T . The disproportionately high value
of the base shears of the 0.40 g 8- or 12-storey frames for DC L is due to
236
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
9
8
8
7
7
7
7
Storey
(a) 9
Storey
9
Exterior
8
Interior
8
6
6
6
6
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
0
9
9
9
9
8
8
0
Storey
Interior
5
1
DC-L
Exterior
Interior
Exterior
Interior
8
8
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
DC-M
0
0
9
9
0
8
8
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
0
9
9
9
8
8
7
7
7
7
0
Exterior
Interior
8
Interior
Interior
8
6
6
6
6
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
EAK
0.0
0
0.2
0.4
0.6
0
0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0
Exterior
Interior
Exterior
Interior
9
Exterior
8
5
0
Interior
0
Exterior
7
DC-H
Exterior
9
9
Exterior
Interior
8
1
Storey
9
Exterior
0.5
1.0
1.5
2.0
Exterior
Interior
0
2.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Figure 2. Beam depth (hb ), beam top (ρ) and bottom (ρ ) reinforcement ratio and column total reinforcement ratio (ρc ) in 8-storey frames for EC8 DC L (1st from top), M
(2nd from top) or H (3rd from top) and for Greek code EAK (bottom). Design PGA:
(a) 0.2 g; (b) 0.4 g. Closed circles: exterior and open circles: interior members.
the high global stiffness and large total mass resulting from the increased
member sizes of these designs.
Table V lists the material quantities required in the alternative frame
designs. The quantities listed are per frame, but include those of the beams
in the transverse horizontal direction (assumed the same as in the beams of
the frame).
237
SEISMIC PERFORMANCE OF RC FRAMES
9
8
8
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
Storey
(b) 9
Storey
1
Storey
Interior
Exterior
8
Interior
8
1
1
1
0
0
0
0
9
9
8
8
7
7
6
5
DC-L
9
Exterior
Interior
Interior
8
7
6
6
6
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
DC-M
0
0
9
9
Exterior
8
8
8
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
0
1
DC-H
0
8
Interior
9
9
8
8
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
1
1
1
2
1
EAK
9
Exterior
Interior
0.0
0.2
0.4
0.6
hb (m)
0
0.8 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0
(%)
Interior
Exterior
Interior
9
Exterior
8
Interior
0
0
Exterior
9
9
Interior
Interior
0
0
Exterior
Exterior
9
Exterior
8
7
1
Storey
9
9
Exterior
0.5
1.0
1.5
' (%)
2.0
8
Exterior
Interior
0
2.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
c (%)
Figure 2. Continued.
3. Evaluation of Frame Performance by Means of Nonlinear Dynamic
Analyses
3.1. Frame modelling for the nonlinear analyses
In the designed buildings member deformation demands due to the design
earthquake are estimated through nonlinear dynamic analysis of the
response to a set of seven motions, emulating historic records from Southern Europe or California and modified to be compatible to the 5%-damped
elastic spectrum of the design earthquake. Figure IV shows the seven
motions and compares their 5%-damped elastic spectra to the (target) Type
238
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
Storey
(a) 13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
DC - L
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1 DC - H
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1 EAK
0
0.0
0.2
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Storey
Storey
Storey
13
12
11
10
9
8
7
6
5
4
3
2
1 DC - M
0
0.4
0.6
hb (m)
0.8
Exterior
Exterior
Interior
Interior
Exterior
Exterior
Interior
Interior
Exterior
Exterior
Interior
Interior
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Exterior
Exterior
Interior
Interior
Exterior
Exterior
Interior
Interior
Exterior
Exterior
Interior
Interior
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Exterior
Exterior
Interior
Interior
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Exterior
Exterior
Interior
Interior
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
13
Exterior 12
12
Interior 11
11
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0.0 0.51.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0
ρ (%)
ρ' (%)
Exterior
Exterior
Interior
Interior
Exterior
Interior
Exterior
Interior
0.5
1.0 1.5
ρc (%)
2.0
2.5
Figure 3. Beam depth (hb ), beam top (ρ) and bottom (ρ ) reinforcement ratio and column total reinforcement ratio (ρc ) in 12-storey frames for EC8 DC L (1st from top),
M (2nd from top) or H (3rd from top) and for Greek code EAK (bottom). Design
PGA: (a) 0.2 g; (b) 0.4 g. Closed circles: exterior and open circles: interior members.
1 spectrum of EC8 for soil type C. Motions and spectra in Figure 4 correspond to a PGA on rock of 1.0 g, i.e., they are scaled to a PGA on soil
C of 1.15 g; in the nonlinear analyses they are applied to the frames scaled
to the design PGA on rock of 0.175 or 0.35 g, i.e., multiplied by a factor
of 0.175 or 0.35 for the 0.20 or 0.4 g designs, respectively.
Mean values of material strengths: fym = 1.15fyk = 575 MPa, fcm =
fck + 8 = 38 MPa are used in the nonlinear analyses and in the evaluation
of member performance on the basis of its results.
239
SEISMIC PERFORMANCE OF RC FRAMES
(b) 13
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Storey
12
11
10
9
8
7
6
5
4
3
2
1
0
DC - L
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1 DC - H
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1 EAK
0
0.0
0.2
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Storey
Storey
Storey
13
12
11
10
9
8
7
6
5
4
3
2
1 DC - M
0
0.4
hb (m)
0.6
0.8
Exterior
Exterior
Interior
Interior
Exterior
Exterior
Interior
Interior
Exterior
Exterior
Interior
Interior
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Exterior
Exterior
Interior
Interior
Exterior
Exterior
Interior
Interior
Exterior
Exterior
Interior
Interior
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Exterior
Exterior
Interior
Interior
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Exterior
Exterior
Interior
Interior
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
13
Exterior 12
12
Interior 11
11
10
10
9
9
8
8
7
7
6
6
5
5
4
4
3
3
2
2
1
1
0
0
0.0 0.51.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0
ρ' (%)
ρ (%)
Exterior
Interior
Exterior
Interior
Exterior
Interior
0.5
1.0
1.5
2.0
2.5
ρc (%)
Figure 3. Continued.
A point-hinge model is used for the members. The yield moment of
the point-hinge is taken equal to the ultimate moment of the section at
exhaustion of the crushing strain of unconfined concrete (0.35%) for the
instantaneous value of the member axial force, as these changes during
the time-history analysis. A simplified Takeda model, as modified by Otani
(1974) and Litton (1975), is used for the hysteresis law between the moment
and the plastic hinge rotation. This law employs: (a) a bilinear skeleton
curve for monotonic loading, (b) unloading to a residual plastic hinge rotation for zero moment equal to 70% of that for fully elastic rotation, and
(c) reloading thereafter towards the extreme previous point on the skeleton
240
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
curve in the direction of reloading. The post-yield hardening ratio is taken
equal to 5%. As hysteretic energy dissipation takes place only after member
yielding, Rayleigh damping is considered with a viscous damping ratio of
5% at the fundamental period of the elastic frame (calculated through the
Rayleigh quotient but using for the elastic structure the member effective
stiffnesses from Eq. 4 below) and at twice that period.
Member elastic stiffness is assumed equal to the member secant stiffness
at yielding of both ends in antisymmetric bending:
EIeff =
My
Ls
3θy
(4)
In Eq. 4 Ls is the shear span, taken here equal to half the member clear
length, while My and θy are the moment and the chord-rotation at yielding of the member end section. The yield moment, My , is taken equal to
the ultimate moment of the section at exhaustion of the crushing strain
of unconfined concrete for the initial value of the axial force (under the
gravity loads which act simultaneously with the design earthquake). The
chord-rotation θy at yielding is estimated through Eq. 4, which accounts
for flexural and shear deformations (1st and 2nd term) and for bond slip
of bars within the joint beyond the end section of the member (3rd term).
θy = ϕy
0.25εy db fy
Ls
+ 0.0025 +
√
3
(d − d ) f c
(5)
In Eq. 5 φy is the yield curvature, computed from first principles, fy
and fc (in MPa) are the strengths of steel and concrete (taken equal to
the mean values), db is the diameter of the tension reinforcement, d, d denote the depth to the tension and compression reinforcement, respectively and εy = fy /Es . Eq. 5 has been fitted by Panagiotakos and Fardis
(2001) to over 1100 test results on RC members with rectangular crosssection.
In beams the effective rigidity given in Eq. 4 is the average of those calculated at the two ends for both positive and negative bending.
It is reminded that chord-rotation is defined as the angle between the
tangent to the axis at the yielding end and the chord connecting that end
to the end of the shear span (point of inflection). It is equal to the member drift ratio, i.e., the deflection of the end of the shear span divided by
Ls . Storey drift ratio is equal to the sum of: (a) the average beam chord
rotation and (b) the average column chord rotation of the storey.
As in the elastic analyses for the design of the frames, columns of the
bottom storey are assumed fixed at grade level; beam gravity loads are
computed on the basis of beam tributary areas in the two-way square slab
system; the finite size of beam–column joints is considered and the joints
4-storey
8-storey
12-storey
Building
0.43
0.81
0.95
693
1404
2839
0.57
0.99
0.92
343
597
1476
Vb
0.55
0.98
1.04
T1
526
912
1996
Vb
0.56
0.79
0.91
T1
1361
2941
5943
Vb
0.4
0.71
0.92
T1
1501
3348
5066
Vb
T1
Vb
T1
0.44
0.93
1.23
T1
769
1319
1886
Vb
EC8 DC H
Greek code
EC8 DC L
EC8 DC H
Greek code
EC8 DC M
Design PGA : 0.40 g
Design PGA : 0.20 g
Table IV. Fundamental period T1 (s) and design base shear Vb (kN) of alternative frame designs
0.47
0.87
1.15
T1
1152
2206
3180
Vb
EC8 DC M
0.39
0.51
0.75
T1
2887
9176
17009
Vb
EC8 DC L
SEISMIC PERFORMANCE OF RC FRAMES
241
2.81
1.22
1.17
4.48
9.25
4.91
3.68
6.29
21.01
10.01
6.66
16.30
16.76
17.64
16.76
19.55
44.54
37.49
37.04
38.07
75.41
74.09
75.41
72.30
124.02
124.02
124.02
124.02
34.80
34.80
34.80
34.80
12.12
12.12
12.12
12.12
concrete
4-storey
EC8 DC L
EC8 DC M
EC8 DC H
Greek code
8-storey
EC8 DC L
EC8 DC M
EC8 DC H
Greek code
12-storey
EC8 DC L
EC8 DC M
EC8 DC H
Greek code
concrete
Beams
and DC
steel
Columns
Design PGA : 0.20 g
Building
7.86
6.63
8.39
9.30
5.55
4.31
4.83
4.64
1.75
1.63
1.79
1.64
steel
199.43
198.11
199.43
196.60
79.34
72.29
71.84
72.87
28.88
29.76
28.88
31.67
concrete
Total
28.87
16.64
15.04
25.60
14.80
9.23
8.52
10.94
4.56
2.85
2.95
6.11
steel
94.15
66.37
67.40
66.37
57.11
45.28
41.67
45.28
22.64
20.58
22.64
20.58
concrete
Beams
22.11
11.56
8.73
16.32
8.08
6.13
4.50
7.85
2.24
1.86
1.39
2.17
steel
Design PGA : 0.40 g
Table V. Concrete volume (m3 ) and steel weight (t) per frame, including transverse beams
175.68
61.39
61.20
63.83
65.88
34.80
34.80
34.97
17.40
17.40
17.40
17.40
concrete
Columns
18.67
9.56
8.84
12.04
7.78
5.20
4.73
6.12
2.76
2.43
2.47
2.24
steel
269.83
127.76
128.60
130.20
122.99
80.08
76.47
80.25
40.04
37.98
40.04
37.98
concrete
Total
40.78
21.12
17.58
28.37
15.86
11.32
9.24
13.97
5.00
4.29
3.86
4.40
steel
242
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
SEISMIC PERFORMANCE OF RC FRAMES
243
are assumed rigid. The contribution of bar pull-out from the joints to the
fixed-end rotation at member ends is considered though, via the 3rd term
in Eq. 5.
In the calculation of the values of My and θy , the width of the slab
which is effective in tension and contributes to the top reinforcement of
the beam end sections with the bars which are parallel to the beam is
taken equal to one-quarter of the beam span on each side of the beam (i.e.
much larger than the conventional effective slab width according to EC 2,
considered in the design). The resulting beam yield moments and flexural
resistances in negative (hogging) bending are much larger than the conventional values taken into account in the design of the frames. This difference promotes flexural plastic hinging in the columns, rather than in beams.
The increased tension reinforcement ratio for negative (hogging) moments
reduces also the ultimate deformation capacity of beams at the supports
(cf. term involving mechanical reinforcement ratios, ω, ω , in Eq. 6 below).
P – effects are considered in the columns.
3.2. Member collapse performance of the frames under the design
earthquake
Frame performance under the design (475 years) earthquake is evaluated on
the basis of the ratio of the chord rotation demand at member ends to
the corresponding supply or capacity. Chord rotation demand is obtained
from the nonlinear dynamic analyses for the seven ground motions. Member chord rotation capacity is taken equal to the chord rotation at which
the member exhibits a drop in peak lateral force resistance during a cycle,
above 20% of the maximum previous lateral resistance during the response
(“loss of lateral load capacity”, considered as a “near-collapse” condition
at the member level). Unlike EC 8, where member deformation capacity is
implicitly quantified on the basis of a simple model of flexural behaviour
(via curvature ductility and the plastic hinge length), in this work an empirical expression is used for member deformation capacity. It has been fitted
by regression to the results of over 1000 monotonic or cyclic tests to failure of flexure-controlled beam-, column- or wall-specimens (Panagiotakos
and Fardis, 2001; fib, 2003). According to this expression, the mean value
of the flexure-controlled ultimate chord rotation of a cyclically loaded RC
member is:
0.2 0.425 f Ls
max(0.01, ω )
αρ yw
θu = αst (0.3 )
fc
25 sx fc ,
max(0.01, ω)
h
v
where
• Ls/h = M/V h: shear span ratio at member end;
(6)
244
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
Kalamata 1986 L
3.5
1.0
3.0
0.5
2.5
A (g)
Acceleration (g)
1.5
0.0
-0.5
0.5
0.0
Friuli 1976 - Tolmezzo X
Acceleration (g)
1.5
3.5
3.0
2.5
1.0
A (g)
0.5
0.0
0.0
Montenegro 1979 - Ulcinj2 X
3.0
2.5
0.5
A (g)
Acceleration (g)
3.5
1.0
0.0
-0.5
2.0
1.5
1.0
-1.0
0.5
-1.5
0.0
Montenegro 1979 - Herceg Novi X
1.5
3.5
3.0
2.5
1.0
0.5
A (g)
Acceleration (g)
1.5
0.5
1.5
0.0
2.0
1.5
1.0
0.5
-0.5
-1.0
0.0
1.5
Loma Prieta 1989 - Capitola 000
3.5
1.0
3.0
2.5
0.5
A (g)
Acceleration (g)
2.0
1.0
-0.5
-1.0
0.0
-0.5
2.0
1.5
-1.0
1.0
0.5
-1.5
0.0
3.5
Imperial Valley 1979 - BondsCorner 140
1.5
1.0
3.0
2.5
0.5
0.0
A (g)
Acceleration (g)
1.5
1.0
-1.0
-1.5
-0.5
2.0
1.5
-1.0
1.0
-1.5
0.5
-2.0
0.0
Imperial Valley 1940 - El Centro Array #9 180
1.5
3.5
1.0
3.0
0.5
2.5
A (g)
Acceleration (g)
2.0
0.0
-0.5
2.0
1.5
1.0
-1.0
0.5
-1.5
0
1
2
3
4
5
6
7
8
9
Time (sec)
10
11 12 13
14 15
0.0
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Period (sec)
Figure 4. Time-histories emulating historic records and their elastic spectra vs. the
target spectrum.
SEISMIC PERFORMANCE OF RC FRAMES
245
• ω, ω : mechanical reinforcement ratios, ρfy/fc , of the tension and compression longitudinal reinforcement (not including any diagonal bars); any
intermediate longitudinal bars between the extreme tension and compression reinforcement are included in the tension steel, ω;
• ν = N/bhc fc : axial load ratio normalized to the width b of the compression zone, the section depth h and fc ;
• ρsx = (Asx /bw sh ): ratio of transverse steel parallel to direction (x) of
loading (sh = stirrup spacing, fyw = stirrup yield stress);
• α: confinement effectiveness factor:
2
bi
1 − sh
1 − sh
1−
(7)
α=
2b0
2h0
6b0 h0
(b0 , h0 = dimensions of confined concrete cores, bi is the distances of
restrained longitudinal bars along the perimeter; cf. footnote (6) in
Table II);
• ast : coefficient for the steel of longitudinal bars, equal to 0.016 for ductile hot-rolled steel and for heat-treated (tempcore) steel, or to 0.0105 for
brittle cold-worked steel;
Eq. 6 includes the effect of the fixed-end rotation at member end due to
pull-out of its reinforcement from the anchorage region beyond the member end.
Due to the large scatter of the fit (coefficient of variation of the ratio of
experimental values to predictions of Eq. 6 of 46%), the 5% fractile of the
ultimate chord rotation is 40% of the mean value given by Eq. 6 and the
mean-minus-one-standard-deviation is about 55% of that value.
The member chord rotation demand-to-supply ratio is considered as a
“damage ratio” against loss of lateral load capacity (“collapse”) of the
member. Figures 5–7 give the range (minimum and maximum) and the
mean value of this “damage ratio” in the beams and columns of the frames
for the seven ground motions, with each motion considered to act in the
positive or in the negative direction.
To assist in the interpretation of the concentration of inelastic deformation and damage in beams or columns, Figure 8 presents the values
of the ratio of the sum of column flexural capacities around joints to
the sum of beam flexural capacities around the same joint, separately
for interior or exterior joints and for each sense of the moments around
the joint. The effect of the fluctuation of axial load on column capacity, which is taken into account during the seismic response analysis, is
neglected in the calculation of column flexural capacity. Beam negative
moment capacities are much larger that in design calculations, due to the
large contribution of slab reinforcement parallel to the beam considered in
the nonlinear response analysis but neglected in design. Despite of that, in
all but the EC8 DC L frames the sum of column capacities exceeds that
246
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
5
(Beams - Exterior)
5
(Columns - Exterior)
4
4
3
3
2
2
2
1
1
4
4
3
3
2
1
5
(Beams - Interior)
Storey
Storey
(a) 5
1
DC - L
0
0
0
5
5
5
4
4
3
3
2
1
(Beams - Interior)
0
5
(Columns - Exterior)
4
4
3
3
2
2
2
1
1
Storey
Storey
DC - L
(Beams - Exterior)
1
0
0
0
0
5
5
5
5
(Columns - Exterior)
(Beams - Interior)
4
3
3
2
4
4
3
3
2
2
2
1
1
Storey
Storey
(Beams - Exterior)
4
0
0
5
5
5
(Beams - Interior)
1
0
5
(Columns - Exterior)
4
4
4
4
3
3
3
3
2
2
2
2
1
1
Storey
Storey
0
1
0.1
0.2
0.3
(Columns - Interior)
1
EAK
EAK
0
0.0
(Columns - Interior)
DC - H
DC - H
(Beams - Exterior)
(Columns - Interior)
DC - M
DC - M
1
(Columns - Interior)
0.4
0
0.0
0.1
0.2
0.3
0.4
0
0.0
0.1
0.2
0.3
0.4
0
0.0
0.1
0.2
0.3
0.4
Figure 5. Minimum–maximum range and mean member “damage ratio” from 7
time-history analyses of 4-storey frames designed to EC8 DC L (1st from top), M
(2nd from top) or H (3rd from top), or to the Greek code EAK (bottom). Design
PGA: (a) 0.2 g; or (b) 0.4 g.
of beams by a factor usually much greater than the factor of 1.3 or 1.4
used in their design to EC8 or the Greek codes, respectively. The columnto-beam-flexural-capacities ratio is usually higher in EC8 DC H designs
than in DC M ones, and higher in EC8 frames than in those according
to the Greek code. This is not due to differences in the value of the overstrength factor of 1.3 or 1.4 employed in these capacity design calculations;
instead, it is an indirect consequence of differences in (beam and column)
minimum reinforcement between these three alternative designs for ductility.
247
SEISMIC PERFORMANCE OF RC FRAMES
5
5
5
(Beams - Exterior)
(Columns - Exterior)
(Beams - Interior)
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
DC - L
DC - L
0
0
0
5
5
5
0
5
(Columns - Exterior)
(Beams - Interior)
4
4
4
3
3
3
3
2
2
2
2
1
1
1
1
DC - M
DC - M
0
0
0
5
5
5
(Beams - Exterior)
(Beams - Interior)
0
5
(Columns - Interior)
(Columns - Exterior)
4
4
4
4
3
3
3
3
2
2
2
2
Storey
Storey
(Columns - Interior)
4
Storey
Storey
(Beams - Exterior)
1
1
1
1
DC - H
DC- H
0
0
0
0
5
5
5
5
(Beams - Interior)
(Beams - Exterior)
(Columns - Interior)
(Columns - Exterior)
4
4
4
4
3
3
3
3
2
2
2
2
1
Storey
Storey
(Columns - Interior)
4
Storey
Storey
(b) 5
1
1
1
EAK
EAK
0
0.0
0
0.1
0.2
0.3
0.4
0
0
0.0
0.1
0.2
0.3
0.4
0.0
0.1
0.2
0.3
0.4
0.0
0.1
0.2
0.3
0.4
Figure 5. Continued.
In DC L frames, which do not have a strong column-weak beam design,
the column-to-beam flexural capacity ratio is around 1.0, often falling
below that value and suggesting plastic hinging in columns.
The main conclusion from Figures 5 to 7 is that, despite their large
differences in member cross-sections and amount and detailing of reinforcement, frames with the same number of storeys designed for the same
PGA but for different ductility have similar performance under the design
earthquake. “Damage ratios” are very consistent between interior and exterior members and fairly similar in frames with different number of storeys.
This means that application of the three alternative ductility options of
248
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
9
(Beams - Exterior)
9
(Beams - Interior)
8
8
8
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
0
0
9
9
9
9
DC - L
DC - L
(Beams - Interior)
8
8
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
0
9
9
1
DC - M
(Beams - Exterior)
DC - M
0
9
(Beams - Interior)
9
(Columns - Exterior)
8
8
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
9
Storey
Storey
(Columns - Interior)
(Columns - Exterior)
8
7
8
DC -H
0
(Beams - Exterior)
8
DC -H
9
9
(Beams - Interior)
0
9
(Columns - Interior)
(Columns - Exterior)
8
8
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0.4 0.0
0
0.0
0.1
0.2
0.3
Storey
8
EAK
0.1
0.2
0.3
0.4
(Columns - Interior)
1
8
0
0.0
(Columns - Interior)
1
8
Storey
Storey
(Beams - Exterior)
Storey
9
(Columns - Exterior)
8
Storey
Storey
(a) 9
1
EAK
0.1
0.2
0.3
0.4
0
0.0
0.1
0.2
0.3
0.4
Figure 6. Minimum–maximum range and mean member “damage ratio” from 7
time-history analyses of 8-storey frames designed to EC8 DC L (1st from top), M
(2nd from top) or H (3rd from top), or to the Greek code EAK (bottom). Design
PGA: (a) 0.2 g; or (b) 0.4 g.
EC 8 results in fairly uniform and consistent performance under the design
earthquake.
More specific conclusions drawn from Figures 5–7 are the following.
– In the frames designed for ductility (i.e., to EC8 DC M or H, or to
the Greek code), the chord rotation demand-to-supply or “damage” ratio
assumes very similar values at all intended plastic hinge locations –
i.e., at ends of beams and at the base of columns at grade level: in
249
SEISMIC PERFORMANCE OF RC FRAMES
(b) 9
8
(Beams - Interior)
8
9
(Columns - Exterior)
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
0
9
9
9
1
DC -L
(Beams - Exterior)
8
(Columns - Exterior)
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0
9
9
(Beams - Exterior)
8
8
Storey
7
DC -M
(Beams - Interior)
8
(Columns - Interior)
8
7
9
1
DC - M
0
9
(Columns - Exterior)
8
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
1
1
DC-H
0
9
Storey
7
1
0
0
9
9
(Beams - Exterior)
Storey
(Beams - Interior)
0
9
7
0
Storey
8
DC - L
7
1
0
9
(Columns - Exterior)
(Beams - Interior)
(Columns - Interior)
8
8
8
7
7
7
7
6
6
6
6
5
5
5
5
4
4
4
4
3
3
3
3
2
2
2
2
1
1
1
0
0.4 0.0
0
EAK
0
0.0
0.1
0.2
0.3
0.1
0.2
0.3
0.4
(Columns - Interior)
2
DC- H
8
Storey
Storey
8
(Columns - Interior)
8
7
Storey
Storey
9
9
(Beams - Exterior)
8
1
EAK
0.0
0.1
0.2
0.3
0
0.4
0.0
0.1
0.2
0.3
0.4
Figure 6. Continued.
general between 0.2 and 0.3 (or 0.1–0.15 at the base of columns of 12storey frames designed for a PGA of 0.2 g). In these frames the value
of the “damage ratio” is much lower over the rest of the column height
than at grade level: around 0.1, or 0.05 in the 12-storey frames designed
for a PGA of 0.2 g. This is consistent with the high column-to-beam
flexural capacity ratios shown in Figure 8 for all these frames. Except
at the base of the frame, no column hinging occurs in these designs.
Beam “damage ratios” decrease slightly when going from EC8 DC H to
DC M, and even further for design to the Greek code. Design to this
latter code gives slightly lower “damage ratio” at the base of columns
250
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0.0
(Beams - Exterior)
DC - M
(Beams - Exterior)
DC- H
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
(Beams - Exterior)
12
11
10
9
8
7
6
5
4
3
2
1
EAK
0
0.1
0.2
0.3
0.4
0.0
Storey
(Beams - Interior)
(Beams - Interior)
Storey
DC- L
13
12
11
10
9
8
7
6
5
4
3
2
1
0
(Beams - Interior)
Storey
Storey
13
12
11
10
9
8
7
6
5
4
3
2
1
0
(Beams - Exterior)
(Beams - Interior)
Storey
12
11
10
9
8
7
6
5
4
3
2
1
0
Storey
Storey
Storey
(a) 13
0.1
0.2
0.3
0.4
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0.0
(Columns - Exterior)
DC - L
(Columns - Exterior)
DC - M
(Columns - Exterior)
DC- H
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
(Columns - Interior)
(Columns - Interior)
(Columns - Interior)
13
(Columns - Interior)
12
11
10
9
8
7
6
5
4
3
2
1
EAK
0
0.3
0.4
0.0
0.1 0.2 0.3
0.4
(Columns - Exterior)
0.1
0.2
Figure 7. Minimum–maximum range and mean member “damage ratio” from 7
time-history analyses of 12-storey frames designed to EC8 DC L (1st from top), M
(2nd from top) or H (3rd from top), or to the Greek code EAK (bottom). Design
PGA: (a) 0.2 g; or (b) 0.4 g.
than design to EC8 DC M or H. Above the base of the columns, the
“damage ratio” is not significantly affected by design to either EC8 DC
M or H, or to the Greek code. Although differences in performance of
the frames designed for ductility are hard to identify, design for EC8
DC H seems to give slightly better overall performance only in the 12storey frame designed for a PGA of 0.4 g. In that particular case the
DC H frame has lower and more consistent “damage ratios” in the
251
SEISMIC PERFORMANCE OF RC FRAMES
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0.0
DC - L
(Beams - Exterior)
DC - M
(Beams - Exterior)
DC - H
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
(Beams - Exterior)
12
11
10
9
8
7
6
5
4
3
2
1
EAK
0
0.1
0.2
0.3
0.4
0.0
(Beams - Interior)
Storey
13
12
11
10
9
8
7
6
5
4
3
2
1
0
(Beams - Interior)
Storey
13
12
11
10
9
8
7
6
5
4
3
2
1
0
(Beams - Exterior)
(Beams - Interior)
Storey
12
11
10
9
8
7
6
5
4
3
2
1
0
(Beams - Interior)
Storey
Storey
Storey
Storey
Storey
(b) 13
0.1
0.2
0.3
0.4
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0.0
(Columns - Exterior)
DC - L
(Columns - Exterior)
DC - M
(Columns - Exterior)
DC - H
0.2
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
0
13
12
11
10
9
8
7
6
5
4
3
2
1
EAK
0
0.3
0.4
0.0
(Columns - Exterior)
0.1
13
12
11
10
9
8
7
6
5
4
3
2
1
0
(Columns - Interior)
(Columns - Interior)
(Columns - Interior)
(Columns - Interior)
0.1
0.2
0.3
0.4
Figure 7. Continued.
columns above the ground storey and overall similar beam performance.
In all other cases, design for EC8 DC M consistently gives slightly superior performance in the beams than design for DC H. As far as the columns are concerned, performance is about the same in all EC8 DC M
or H frames, except for the 12-storey ones designed for a PGA of 0.4 g.
– Frames designed to EC8 DC L (i.e., for strength alone rather than
for controlled inelastic response and ductility) have always lower beam
“damage ratios” than EC8 DC M or H frames, despite (a) the less
confining reinforcement and the lower minimum compression reinforcement ratio of DC L beams and (b) the reduction in beam deformation
252
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
capacity effected by the smaller shear span ratio, Ls /h, of DC L beams
which are deeper than their DC H or M counterparts. This is attributed to: (a) the large flexural capacity of these beams, which promotes
plastic hinging and inelastic deformations in columns instead of beams
(cf. Figure 8); (b) the increase in beam deformation capacity due to the
heavy bottom reinforcement owing to proportioning of the end sections
of these beams for the large positive seismic moments of the analysis for
the design seismic action. The columns of all DC L frames designed for
a PGA of 0.2 g, as well as those of the 4-storey frames designed for a
PGA of 0.4 g, have similar damage ratios as in the frames designed for
ductility. This happens despite the apparent occurrence of inelastic deformations at several levels of the columns of the DC L frames and the
effect of the lower confining reinforcement on their deformation capacity; it is attributed to the reduction in deformation demands owing to the
larger overall effective stiffness of DC L frames due to their heavier reinforcement. Given that DC L frames designed for a PGA of 0.2 g, as well
as DC L 4-storey frames designed for a PGA of 0.4 g have also slightly
superior beam performance, their overall performance under the design
earthquake is equal or slightly better than that of the frames designed
for ductility.
– The columns of 8- or 12-storey EC8 DC L frames designed for a PGA
of 0.4 g do not show a clear concentration of plastic hinge rotations at
the base. This is not surprising, in view of the column-to-beam capacity ratios in Figure 8. Moreover, they show a very large scatter of the
“damage ratio” in the upper storeys, indicative of erratic column hinging due to higher-mode response. “Damage ratios” are large in the upper
storey columns of these designs, despite: (a) the reduced deformation
demands due to the larger overall effective stiffness of these frames,
effected by the large cross-section and the heavy reinforcement of their
members; (b) the increase in column chord-rotation capacity due to the
lower column axial load ratio, νd , effected by the larger column size
(cf. Table I). It is concluded, therefore, that for a design PGA of 0.4 g,
in addition to being not cost-effective, design to EC8 DC L gives also
inferior overall performance under the design earthquake.
– The maximum values of the “damage ratio” for the seven time-histories
are always less than the threshold of 0.4 corresponding to the 5% fractile
of the capacity; therefore a margin exists against the inherent uncertainty
of the ultimate deformation capacity.
The satisfactory performance of the frames under the design earthquake
is noteworthy, in view of the fact that performance is evaluated here on the
basis of displacements and deformation demands computed using (in the
nonlinear dynamic analyses) member elastic stiffness equal to the member
253
SEISMIC PERFORMANCE OF RC FRAMES
(a) 5
exterior
interior
exterior
interior
DCL
DCM
exterior
interior
interior
exterior
4
Storey
3
2
1
EAK
DCH
0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0
9
8
exterior
interior
exteriorr
interior
DCL
DCM
5.0 6.0 0.0 1.0 2.0
exterior
interior
3.0 4.0
5.0 6.0
interior
exterior
7
Storey
6
5
4
3
2
1
EAK
DCH
0
0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0 3.0 4.0
13
12
exterior
interior
exterior
interior
5.0 6.0 0.0 1.0 2.0
exterior
in
ri
interior
3.0 4.0
5.0 6.0
ri
interior
exterior
11
10
9
Storey
8
7
6
5
4
3
2
1
0
DCL
0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0
2.0 3.0 4.0
DCM
DCH
5.0 6.0 0.0 1.0 2.0 3.0 4.0
5.0 6.0 0.0 1.0 2.0
EAK
3.0 4.0
5.0 6.0
Figure 8. Ratio of sum of column flexural capacities to sum of beam flexural capacities around joints in frames designed to EC8 DC L (1st from left), M (2nd from left),
or H (3rd from left), or to the Greek code EAK (right). Design PGA: (a) 0.2 g; (b)
0.4 g. Closed circles: exterior joints; open circles: interior joints.
secant stiffness at yield, which is much lower than the conventional member stiffness used in the design.
3.3. Damage limitation performance of the frames under the damage
limitation earthquake
Frame members were sized so that the interstorey drift ratio under the
damage limitation earthquake (taken as 50% of the design earthquake in
254
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
(b) 5
exterior
interior
exterior
interior
interior
exterior
exterior
interior
4
Storey
3
2
1
DCM
DCL
0
0.0
9
1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0
3.0
exterior
interior
exterior
interior
8
4.0 5.0 6.0 0.0 1.0
EAK
DCH
2.0 3.0 4.0
5.0 6.0 0.0 1.0
2.0
3.0
4.0
5.0 6.0
interior
exterior
exterior
interior
7
Storey
6
5
4
3
2
1
DCM
DCL
EAK
DCH
0
0.0
13
12
1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0
3.0
4.0 5.0 6.0 0.0 1.0
exterior
interior
exterior
interior
2.0 3.0 4.0
5.0 6.00.0 1.0
2.0
3.0
4.0
5.0 6.0
interior
exterior
exterior
interior
11
10
9
Storey
8
7
6
5
4
3
2
1
0
DCM
DCL
0.0 1.0 2.0 3.0 4.0 5.0 6.0 0.0 1.0 2.0
3.0
4.0 5.0 6.0 0.0 1.0
EAK
DCH
2.0 3.0 4.0
5.0 6.0 0.0 1.0
2.0
3.0
4.0
5.0 6.0
Figure 8. Continued.
EC8 or 40% of it in the Greek code), nowhere exceeds the damage limitation threshold of 0.5%. Frame performance under 50% of the design
earthquake is evaluated via nonlinear dynamic analysis of the response to
the same set of seven motions used in the analyses for the design earthquake, scaled this time to the PGA of 50% of the design earthquake.
(For comparison between designs, the fact that in the design to the Greek
code the damage limitation earthquake is 40% of the design earthquake is
neglected.) The same modelling is used as in the response analyses for the
design earthquake.
255
SEISMIC PERFORMANCE OF RC FRAMES
(a) 5
4
Storey
3
2
1
DC - L
DC - M
DC - H
EAK
DC - L
DC - M
DC - H
EAK
0
9
8
7
Storey
6
5
4
3
2
1
0
13
12
11
10
9
Storey
8
7
6
5
4
3
2
1
0
0.00
DC - M
DC - L
0.50
1.00
1.50 0.00
0.50
1.00
1.50 0.00
EAK
DC - H
0.50
1.00
1.50 0.00
0.50
1.00
1.50
Figure 9. Minimum–maximum range and mean storey drift ratio from 7 time-history
analyses under the damage limitation earthquake for design PGA: (a) 0.2 g; or (b)
0.4 g. Top: 4-storey frames; middle: 8-storey frames; bottom: 12-storey frames.
Figure 9 shows the range (minimum and maximum) and the mean value
of the computed storey drift ratios over the seven input ground motions
scaled to 50% of the design earthquake. Regardless of the design, drifts
consistently decrease with increasing number of storeys. The heightwise distribution of drifts is fairly uniform, with the exception of the ground storey where the assumed column fixity at grade level reduces the interstorey
drift. There is a clear effect of higher modes on the upper storey drifts in
the more flexible 12-storey frames designed for ductility (i.e., in all designs
except DC L). Among the alternative designs, the lower the q-factor for
256
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
(b) 5
4
Storey
3
2
1
DC - L
DC - M
DC - H
EAK
DC - L
DC - M
DC - H
EAK
0
9
8
7
Storey
6
5
4
3
2
1
0
13
12
11
10
9
Storey
8
7
6
5
4
3
2
1
0
0.00
DC - M
DC - L
0.50
1.00
1.50 0.00
0.50
1.00
1.50 0.00
DC - H
0.50
1.00
1.50 0.00
EAK
0.50
1.00
1.50
Figure 9. Continued.
which the frame has been designed, the lower in general the drifts. This is
due to the increase in overall effective stiffness with the heavier longitudinal reinforcement resulting from the lower q-factor. The difference is largest
between the three alternative designs for ductility (which, in general, have
similar drifts) and the EC8 DC L frames, which are designed with q = 1.5
(moreover, the 0.4 g PGA DC L designs have larger member cross-sections
than the other three designs).
With the exception of the 12-storey frames designed for a PGA of 0.2 g,
the drifts of all frames designed for ductility (i.e., of EC8 DC M or H,
and of the Greek code designs) exceed the damage limitation threshold
of 0.5%. The main reason is that the member elastic stiffness from Eq. 4
used in the nonlinear seismic response analysis is, on average, about 25%
SEISMIC PERFORMANCE OF RC FRAMES
257
of the stiffness of the uncracked gross section, i.e., about half of the conventional value used in design for the storey drift limitation. Plastic hinging
and P − effects increase computed drifts further. Given all these sources
of deviation, the magnitude of the violation of the 0.5% threshold is still
small. Except in the 4-storey frames, EC8 DC L designs meet the 0.5%
drift of the code.
4. Conclusions: Seismic Performance and Cost Effectiveness of Alternative
Designs
For the first time after the finalisation of the European Norm for seismic design of buildings (EC8), the performance of RC buildings designed
with this code is evaluated through systematic nonlinear analyses. Regular
4-, 8- or 12-storey RC frames, designed for a PGA of 0.2 or 0.4 g and to
one of the three alternative ductility classes in EC8, have been evaluated
at the member collapse prevention performance level under their design
earthquake and at the damage limitation level under 50% of the design
earthquake. As the Eurocodes are meant to replace soon existing national
codes, design and performance is also compared to that of similar frames
designed with the 2000 Greek national codes.
Member collapse prevention performance is evaluated through the ratio
of chord rotation demand at member ends (from nonlinear dynamic analyses for seven ground motions compatible with the 5%-damped elastic
response spectrum of the design seismic motion), to the corresponding ultimate cyclic chord rotation capacity, estimated from an empirical expression fitted to a very large number of test results on RC members. This
demand-capacity-ratio may be interpreted as “damage ratio” of the members against loss of lateral load capacity and collapse.
On the basis of the computed values of the member “damage ratio”,
design for ductility according to either Ductility Class (DC) Medium (M)
or High (H) of EC8, or to the 2000 Greek codes is found quite effective in
meeting the collapse prevention objective of Eurocode 8 and of the Greek
codes. More specifically:
• Columns are found to essentially remain elastic above the base and
develop there “damage ratios” around 10%. Capacity design of columns in
flexure for a strong column/weak beam frame provides with sufficient margin to overcome the increase in beam negative moment capacity due to the
contribution of slab reinforcement parallel to the beam, and prevent plastic
hinging in columns instead of beams.
• “Damage ratios” at intended plastic hinge locations – i.e., at ends of
beams and at the base of the columns – are relatively low: between 20 and
35%, well below the threshold value of 40% that corresponds to exceedance
of the 5% fractile of member deformation capacity there.
258
TELEMACHOS B. PANAGIOTAKOS AND MICHAEL N. FARDIS
• “Damage ratios” are very consistent between interior and exterior beams
or columns and fairly uniform between storeys above the ground storey.
• Overall the performance of frames of EC8 DC M or H and of frames
designed to the Greek codes is similar. In beams “damage ratios” decrease
slightly with increasing amount of reinforcement: i.e., from DC H to DC
M to the Greek code.
• Interstorey drift ratios and “damage ratios” are fairly similar in buildings
with 4, 8 or 12 storeys; in general both decrease with increasing number of
storeys.
The damage limitation check (control of storey drift ratio to an upper
limit of 0.5% for brittle nonstructural infills in contact with the RC frame)
under a damage limitation earthquake taken as 50% of the design earthquake, plays a major role for meeting the member collapse prevention
objective under the design earthquake. This check, verified in design using
the conventional stiffness of 50% of the uncracked gross section stiffness,
results in large column sizes that often satisfy the strong column/weak
beam rule with the minimum reinforcement and the verification in flexure
for the Ultimate Limit State.
The performance of all frames (even of those not designed for ductility) is satisfactory under the design (475 years) earthquake and acceptable under the damage limitation one. Differences in seismic performance of the four alternative designs are limited, in view of their large
difference in material quantities. Design for strength instead of ductility
(as in EC8 DC L):
• Gives significantly inferior performance under the design earthquake, for
medium-high rise frames in high seismicity.
• Gives slightly better performance under the design earthquake in moderate seismicity, or even in high seismicity but for low-rise frames.
• Gives better performance at the damage limitation earthquake.
• Requires much larger material quantities and therefore – in view of the
fairly similar performance of the four alternative designs – is much less
cost-effective than design for ductility; for medium-high rise frames, it is
rarely a viable option even in moderate seismicity (PGA of 0.2 g).
Therefore, the recommendation in EC8 to limit the application of DC L
to low seismicity regions (suggested as those with a design PGA of 0.1 g or
less), although not supported by the present results on safety and performance grounds, seems fully justified on the basis of cost-effectiveness and
soundness of the design.
Between the two alternatives of design for ductility in EC8, DC M
seems to be slightly more cost-effective for low-rise frames in moderate
seismicity. DC H is more cost-effective in high seismicity, as well as for
medium-high rise buildings in moderate seismicity. Both these two ductility classes are far more cost-effective than the Greek codes of 2000: frames
SEISMIC PERFORMANCE OF RC FRAMES
259
designed to these latter codes have a marginally better performance than
those designed to EC8 DC M or H; the difference in performance, though,
is imperceptible, compared to the much larger quantity of steel required by
the Greek codes, especially for medium-high rise buildings.
References
CEN (2003). prEN1998-1:2003 Eurocode 8: Design of structures for earthquake resistance.
Part 1: General rules, seismic actions and rules for buildings. Formal Vote Version (Stage
49), November 2003.
Fédération internationale du béton (fib) (2003) Seismic assessment and retrofit of RC
buildings. Chapter 4: fib Bulletin No. 24, Lausanne.
Litton, R.W. (1975). A Contribution to the Analysis of Concrete Structures under Cyclic
Loading. Ph.D. Thesis. Civil Engineering Dept., Univ. of California, Berkeley, California.
Otani, S. (1974) Inelastic Analysis of R/C Frame Structures. ASCE, Journal of Structural
Division 100 (ST7), 1433–1449.
Panagiotakos T. and Fardis M.N. (2001). Deformations of R.C. members at yielding and
ultimate. ACI Structural Journal 98(2) 135–148.
Vidic, T., Fajfar, P. and Fischinger, M. (1994) Consistent inelastic design spectra: Strength
and displacement. Earthquake Engineering and Structural Dynamics 23 502–521.