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OS MODELER - EXAMPLES OF APPLICATION
Version 1.0
(Draft)
Matjaž Dolšek
February 2008
Content
1. Introduction .................................................................................................................................... 1 2. Four-storey reinforced concrete frame designed according to EC8 ............................................... 2
2.1. Description of the structure.................................................................................................... 2 2.2. Pseudo-dynamic tests on full scale specimen ........................................................................ 4 2.3. Mathematical modeling ......................................................................................................... 6
2.3.1. Gravity loads ................................................................................................................. 7 2.3.2. Beam effective width .................................................................................................... 8 2.3.3. Moment-rotation relationship for plastic hinges ......................................................... 11
2.4. Eigen value analysis............................................................................................................. 18 2.5. Pushover analysis................................................................................................................. 18 2.6. Nonlinear dynamic analysis and comparison with pseudo-dynamic test............................. 23 2.7. Incremental dynamic analysis.............................................................................................. 23 2.8. Determination of the target displacement by N2 method .................................................... 28 2.9. Determination of the target displacement by SDOF-IDA approach .................................... 34 2.10. Influence of some modeling uncertainty on the response of structure................................. 35
3. ICONS frame................................................................................................................................ 40 3.1. Description of the structure.................................................................................................. 40 3.2. Pseudo-dynamic tests........................................................................................................... 40 3.3. Mathematical modeling ....................................................................................................... 42 3.4. Pushover analysis for model H and model T ....................................................................... 43 3.5. Nonlinear dynamic analysis and comparison with pseudo-dynamic test............................. 46
4. References .................................................................................................................................... 49
1. Introduction
The objective of this report is to show the capability of the OS Modeler [Dolšek, 2008], which consists
of a set of Matlab functions used for determination of the structural model, for performing nonlinear
analyses by employing program OpenSees [PEER, 2007] and for post-processing of the analysis
results.
In the first example the pushover analysis, nonlinear dynamic analysis and IDA analysis is presented
for the four-storey reinforced concrete frame building, which has been designed according to previous
versions of Eurocodes 2 and 8 [Fardis (Ed.), 1996]. The results of the nonlinear dynamic analyses are
compared with the results of the pseudo-dynamic tests, which were performed in the ELSA
Laboratory in Ispra [Negro and Verzeletti, 1996, Negro et al., 1996]. The results of the pushover
analysis are used to determine the target displacement for a defined seismic loading. The target
displacement is determined according to the N2 method [Fajfar, 2000] and according to the SDOF-
IDA approach. Further on the influence of some selected modeling uncertainties on the seismic
response is presented.
The four-storey frame, designed to reproduce the design practice in European countries about forty to
fifty years ago [Carvalho and Coelho (Eds.), 2001], is the second presented example. In this case only
the pushover analysis and the nonlinear dynamic analysis are performed. The reuslts of the nonlinear
dynamic analysis are presented with the results of the pseudo-dynamic test [Carvalho and Coelho
(Eds.), 2001].
1
2. Four-storey reinforced concrete frame designed according to EC8
2.1. Description of the structure
The elevation and plan of the four-storey reinforced concrete building, as well as the typical
reinforcement of columns and beams are shown in Figure 2.1. The height of the bottom storey is 3.5
m. In other stories the height is reduced for 0.5 m. The building has two bays in each direction with
the raster of 5 m in the X direction and with the raster of 4 and 6 m in the Y direction. This direction
was also the direction of loading in the pseudo-dynamic test. The columns are 40/40 cm. The only
exception is the column D (Figure 2.1), which is 45/45 cm. All beams have rectangular cross section
with 30 cm width and 45 cm height. The slab has the thickness of 15 cm.
Figure 2.1. The four-storey reinforced concrete frame building.
The concrete C25/30 is used for this building and the B500 Tempcore reinforcing steel for which the
characteristic yield strength is 500 MPa. Since the pseudo-dynamic test was performed for the studied
building more information regarding material characteristics is available. In Table 2.1 the mean
concrete strength and modulus of elasticity is presented. The mean concrete strength differs from 32
MPa to 56 MPa. The smallest strength corresponds to columns in third storey and the highest concrete
strength corresponds to beams in first storey. Similarly, the modulus of elasticity varies from 28.5 GPa
to 35.3 GPa. It should be emphasized that the material characteristics of concrete significantly differs
2
from the nominal material characteristics for C25/30, which are, according to Eurocode 2 [CEN,
2004], 33 MPa for mean concrete strength and 31 GPa for modulus of elasticity. The yield strength,
ultimate strength and the corresponding deformation of reinforcing bars are presented in Table 2.2.
The mean yield strength exceeds the characteristic yield strength for 10 to 20%, depending on the
diameter of the reinforcing bar.
The structure was designed according to previous versions of Eurocodes 2 and 8 [Fardis (ed.), 1996].
The design spectrum was defined based on the prescribed peak ground acceleration of 0.3 g, the soil
type B, the ductility class high (DCH) and the behavior factor q=5. In addition to the self weight of the
structure the 2 kN/m2 of permanent load was assumed in order to represent floor finishing and
partitions, and 2 kN/m2 of live load was also adopted. The corresponded masses are 87 t, 86 t and 83 t
for bottom, second and third, and top storey, respectively. These masses were adopted also in the
pseudo-dynamic test on the full scale specimen. The design base shear versus the weight of the
structure corresponded to about 16%, since the design base shear was 529 kN [Fardis (ed.), 1996].
The longitudinal and shear reinforcement of beams and columns is presented in Figures 2.1, 2.2 and
2.3. The reinforcement is presented only for the frames which are parallel to the direction of loading in
the pseudo-dynamic test (Figure 2.1).
Columns are reinforced with 8 to 12 bars with the total reinforcement ratio from 1% to 1.9% as
presented in Figure 2.1. The diameters used for longitudinal reinforcement in columns vary from φ16
at upper stories and φ20 or φ25 in the bottom and in some cases in the first storey. Stirrups φ10/10 are
usually used in the critical regions of columns (Figures 2.1 and 2.3).
Table 2.1. The mean concrete strength and modulus of elasticity.
Construction element Storey
Concrete strength fcm
(MPa)
Modulus of elasticity Ecm
(MPa) 1 49.8 33700 2 47.6 33200 3 32.0 28500
Column
4 46.3 32800 1 56.4 35300 2 53.2 34600 3 47.2 33100
Beam
4 42.1 31700
3
Table 2.2. The mean yield strength of reinforcement bars.
Bar diameter
(mm)
Yield strength (MPa)
Ultimate strength (MPa)
Ultimate deformation
(%) 6 566.0 633.5 23.5 8 572.6 636.1 22.3
10 545.5 618.8 27.5 12 589.7 689.4 23.0 14 583.2 667.4 22.7 16 595.7 681.0 20.6 20 553.5 660.0 23.1 25 555.6 657.3 21.6
Figure 2.2. The longitudinal and shear reinforcement in columns.
Bars φ14 in combination with bar φ12 are used for the longitudinal reinforcement in beams. In all
cases more than half longitudinal reinforcement in the beams is placed at the bottom of the beam
(Figures 2.1 and 2.2). The detailed data of lap-splicing and anchorage of bars is not available.
2.2. Pseudo-dynamic tests on full scale specimen
Different pseudo-dynamic tests were performed for the studied building at the European Laboratory
for Structural Assessment (ELSA, Ispra) [Negro and Verzeletti, 1996, Negro et al., 1996]. The pseudo-
dynamic tests were performed not only for the bare frame but also for different configurations of the
masonry infills. For example, the uniformly infilled bare frame is presented in Figure 2.4.
4
Figure 2.3. The longitudinal and shear reinforcement in beams.
The so called low- and high level tests were performed on the same bare frame. The accelerogram
used in the test is generated from the real accelerogram recorded during the 1976 Friuli Earthquake.
The accelerogram and the corresponding spectrum are presented in Figure 2.5. It is shown that the
acceleration spectrum shape approximately corresponds to the EC 8 shape of spectrum, normalized to
peak ground acceleration of 0.3 g. The scale factors 0.4 (0.12 g) and 1.5 (0.45 g) for the acceleorgram
were used for the low- and high- test, respectively, and the zero viscous damping was assumed in both
tests.
After the low-level test no visible damage were observed. It was assumed that structure become
practically in the elastic region. During the high-level test crakes opened and closed in the critical
regions of the beams of the first three stories and of most columns. Neither spalling of the concrete
cover nor local buckling of reinforcement was observed. Besides the cracks at the end of beams and
columns, which were considered as evidence of yielding in the rebars and of bond-slip in the joints,
the specimen remained quite undamaged. However, the fundamental period of the building after the
high level test was about 1.22, which is about two times higher than the period measured on the
undamaged building (0.56 s).
Quite uniform damage pattern was observed in both test, with exception of fourth storey, for which the
drifts are significantly lower than these measured in other stories. The maximum top displacement in
5
Figure 2.4. The tested frame with masonry infills.
Figure 2.5. The accelerogram used in the pseudo-dynamic test and the corresponding elastic
acceleration spectrum compared with EC8 spectrum.
2.3. Mathematical modeling
The mathematical model of the studied building is developed on basis of Eurocode 8 [CEN, 2004]
requirements. According to Eurocode 8 a bilinear force–deformation relationship may be used at the
element level as the minimum representation of non-linear behaviour of structural elements. In ductile
elements, expected to exhibit post-yield excursions during the response, the elastic stiffness of a
bilinear relation should be the secant stiffness to the yield-point. The trilinear force-deformation
relationships, which take into account pre-crack and post-crack stiffnesses, are allowed. Also, zero
post-yield stiffness may be assumed. However, for elements in which the strength degradation is
expected, the strength degradation has to be included in the force-deformation relationship. It is also
recommended to use the mean values of the properties of the materials. For new structures, mean
values of material properties may be estimated from the corresponding characteristic values. Gravity
6
loads has to be applied to appropriate elements of the mathematical models. Therefore the axial forces
due to gravity loads should be taken into account when determining force-deformation relations for
structural elements.
The same basic principles of the modeling as presented by Fajfar et al. [2006] were employed also in
this study. These principles are full compliance with Eurocode 8. Beam and column flexural behavior
was modeled by one-component lumped plasticity elements, composed of an elastic beam and two
inelastic rotational hinges (defined by the moment-rotation relationship). The element formulation was
based on the assumption of an inflexion point at the midpoint of the element. For beams, the plastic
hinge was used for major axis bending only. Additional plastic hinges in beams were modeled in
series after the primary plastic hinge in order to simulate bar slip, which was observed in the
experiment. For columns, two independent plastic hinges for bending about the two principal axes
were used.
All analyses were performed by OpenSees [PEER, 2007], which is an object-oriented software
framework for simulation applications in earthquake engineering using finite element method,
developed at the Pacific Earthquake Engineering Research Center. The tcl input files for the OpenSees
were automatically generated with the OS modeler developed in Matlab [Dolšek, 2008]. Such
approach has an advantage, since all the moment-rotation envelopes of plastic hinges can be
automatically generated, which is not an option in OpenSees, when plastic hinges are represented by
zero length elements.
In the next Sections more precise description of mathematical modeling, especially, the procedure for
determination of moment-rotation envelopes for plastic hinges, is presented.
2.3.1. Gravity loads
Gravity load was modeled as a uniformly distributed load on beams and as point loads on columns.
The uniformly distributed load on beams results from the self weight of slab and beams and also from
the permanent load on slab. The point loads at top of the columns are used to model only the self
weight of columns.
The specific weight of the reinforced concrete 25 kN/m3 was adopted. Since the gravity load slightly
exceeds the weight calculated from the mass, which was assumed in the pseudo-dynamic experiment,
the live load 2 kN/m2, which was used for the design purpose, has been slightly decreased in order to
obtain the same gravity load as result from the storey mass used in the pseudo-dynamic test. Values
7
for uniformly distributed gravity load as well as point load on columns are presented in Tables 2.3 and
2.4, respectively.
Table 2.3. The uniformly distributed gravity load on beams.
Beam Storey 1
g (kN/m) Storey 2
g (kN/m) Storey 3
g (kN/m) Storey 4
g (kN/m)
1 9.36 9.36 9.36 9.64
2 9.36 9.36 9.36 9.64
3 10.68 10.68 10.68 11.03
4 15.49 15.49 15.49 16.09
5 10.68 10.68 10.68 11.03
6 15.49 15.49 15.49 16.09
7 15.49 15.49 15.49 16.09
8 10.68 10.68 10.68 11.03
9 15.49 15.49 15.49 16.09
10 10.68 10.68 10.68 11.03
11 12.00 12.00 12.00 12.41
12 12.00 12.00 12.00 12.41
Table 2.4. The point loads at the top of columns.
Column Storey 1 G (kN)
Storey 2 G (kN)
Storey 3 G (kN)
Storey 4 G (kN)
1 13.0 12.0 12.0 6.0
2 13.0 12.0 12.0 6.0
3 13.0 12.0 12.0 6.0
4 13.0 12.0 12.0 6.0
5 16.5 15.2 15.2 7.6
6 13.0 12.0 12.0 6.0
7 13.0 12.0 12.0 6.0
8 13.0 12.0 12.0 6.0
9 13.0 12.0 12.0 6.0
2.3.2. Beam effective width
Reinforced concrete beams and the slab act as a monolithic section since they are usually constructed
at the same time. Therefore a contribution of slab to the stiffness and strength of beam has to be
considered in the seismic assessment, although there are no straight rules how to define the beam
8
effective width. Different beam effective width can be used for modeling stiffness and strength
especially in the design process. According to Paulay and Priestley [1992] the flange contribution to
stiffness in T and L beams is typically less than the contribution to flexural strength. They suggested
that the contribution of slab to the beam effective width for modeling of the beam stiffness should be
half of that used for modeling the flexural strength of the beam if the flange is in compression. These
values for beam effective width are presented in Table 2.5. However, different values are used for the
effective tension reinforcement. Paulay and Pristeley [1992] recommended that in T and L beams,
built integrally with floor slabs, the longitudinal slab reinforcement placed parallel with the beam, to
be considered effective in participating as beam tension (top) reinforcement, in addition to bar placed
within the web width of the beam. The tension reinforcement should include all bars within the
effective width beff, which may be assumed to be the smallest of the following
one-fourth of the span of the beam under consideration (lb/4), extending each side from the
center of the beam section, where a flange exists
one-half of the span of a slab, transverse to the beam under consideration, extending each side
from the center of the beam section where a flange exists
one-forth of the span length of a transverse edge beam, extending each side of the center of the
section of that beam which frames into an exterior column an is thus perpendicular to the edge
of the floor.
Table 2.5. The beam effective width according to Paulay and Pristley [1992].
Flexural strength (flange in compression) Flexural stiffness
beff ≤ beff ≤ bw+16hs bw+8hs
bw+(beff,1+beff,2)/2 bw+(beff,1+beff,2)/4 bb/4 bb/8
Figure 2.6. The beam effective width.
9
According to FEMA 356 [2000] the combined stiffness and strength for flexural and axial loading
shall be calculated considering a width of effective flange on each side of the web equal to the smaller
of
the provided flange width
eight times the flange thickness
half the distance to the next web
one-fifth of the span for beams.
When the flange is in compression, both the concrete and reinforcement within the effective width
shall be considered effective in resisting flexure and axial load. When the flange is in tension,
longitudinal reinforcement within the effective width and that is developed beyond the critical section
shall be considered fully effective for resisting flexural and axial loads.
The Eurocode 8 [CEN, 2004] also prescribes that slab reinforcement parallel to the beam and within
the effective flange width, should be assumed in the design process to contribute to the beam flexural
capacities, if it is anchored beyond the beam section at the face of the joint. It is suggested that for
primary seismic beams framing into exterior columns, the effective flange width beff is taken, in the
absence of a transverse beam, as being equal to the width of the column, or, if there is a transverse
beam of similar depth, equal to this width increased by 2hs on each side of the beam.
Also Eurocode 2 [CEN, 2004] suggests values of beam effective widths for all limit states (strength
and stiffness) and are to be based on the distance l0 between points of zero moments
,eff eff i wb b b b= + ≤∑ (0.1)
, 00, 2 0,1 0,2eff i ib b l 0l= ⋅ + ⋅ ≤ ⋅ (0.2)
,eff i ib b≤ (0.3)
where beff is the beam effective width, bw is the width of the beam, bi is the one half of the distance
between the beams and l0 is the distance between the points of zero moments. The distance l0 should
be taken as lb/2 in the seismic analysis.
More extensive review regarding the beam effective width can be found in Stratan and Fajfar [2002].
In the mathematical model used in this case study the beam effective width was calculated according
to the Eurocode 2. The results are presented in the Table 2.6. The beam effective width is the highest
for the beams connected to the interior column and is in the range from 110 to 150 cm. For the exterior
beams the beam effective width is in the range between 70 and 90 cm.
10
Table 2.6. The beam effective width considered in analysis.
Beam beff, EC2 (cm) B3, B5 70
B1, B2, B11, B12 80 B8, B10 90
B4 110 B6, B7 130
B9 150
2.3.3. Moment-rotation relationship for plastic hinges
The only nonlinear behavior of the columns and beams was modeled by moment-rotation plastic
hinges, which were considered at both ends of structural member. It was assumed that the moment-
rotation relationship is tri-linear with the material softening after the maximum moment (Figure 2.7).
The three points at the increasing part of the moment-rotation envelope represent the points at the
cracking of the concrete (CR), the point at the yielding of reinforcement (Y) and the point at maximum
moment (M). After the maximum moment is attained the linear strength degradation is assumed and
defined with the point at the near collapse (NC).
Figure 2.7. Moment-rotation relationship for plastic hinges in beams and columns.
Since the plastic hinges were modeled with the ZeroLengthSection elements, implemented in
OpenSees [PEER, 2007], only the plastic rotation are used in the definition of the moment-rotation
relationship.
11
Cracking (Mcr), yield (My) maximum (Mm) moment and moment at near collapse limit state (Mnc)
The moment at the cracking of concrete is determined based on the elastic analysis of the cross-section
according to the following equation
cr ctPM W fA
⎛= −⎜⎝ ⎠
⎞⎟ (0.4)
where W is section modulus, P is the axial force (compression is negative), A is cross-section area and
the fct is the mean tensile strength determined according to EC 2
( )2 30.3ctm cmf f= (0.5)
where fcm is the mean concrete compressive strength.
The yield and the maximum moment were calculated based on the moment-curvature analysis of the
cross-section. Axial forces due to the vertical loading were taken into account in columns, while in the
beams zero axial force was assumed. For the concrete the stress-strain relationship prescribed by EC 2
for nonlinear analysis was adopted, while for the steel the elasto-plastic strain-stress relationship was
assumed in the analysis.
The yield moment My is defined when the strain in the first reinforcing bar is equal to the yield strain
of the steel fsy/Es, where fsy is the yield stress of the steel and Es the corresponding modulus of
elasticity, which is assumed 21000 kN/cm2. The maximum moment Mm is directly determined from
the moment-curvature analysis. The moment-curvature analysis was determined until the ultimate
deformation of concrete εcu=-3.5 ‰ or the ultimate deformation of reinforcing steel εsu=10 ‰ is
reached.
In some cases it may happened that the section collapse before the yield deformation is reached in the
reinforcing bar. In this case the yield moment is determined based from the assumption of equal areas,
which are determined from the moment-curvature envelope and the idealized Mcr and My points.
The moment at the near collapse limit state Mnc is defined according to EC 8 as the 0.8Mm.
Rotation at Mcr and My
The hinge rotation corresponding to the Mcr is calculated by assuming the linear curvature along the
length of the element
12
0
3cr
crM L
EIΘ = (0.6)
where the Mcr is the cracking moment, L0 is the length between the end of the element and the zero
moment point, in our case is for beams and columns assumed half of the element length, and the EI is
the product of the modulus of elasticity an moment of inertia. Since only the plastic rotation is needed
to define in the plastic hinge, the rotation according to Eq. (0.6) is used to define the elastic rotation at
the plastic hinge. Of course, the elastic rotation increases or decreases if the moment increases or
decreases.
The yield rotation of plastic hinge is determined according to Fischinger [1989]
2
0 1 26
cr cr cry cr
y y y
L M M MM M M yφ φ
⎛ ⎞⎛ ⎞⎛ ⎞ ⎛ ⎞⎜ ⎟⎜Θ = + + − −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠⎝ ⎠⎝ ⎠
⎟ (0.7)
where L0 is the length between the end of the element and the zero moment point, in our case is for
beams and columns assumed half of the element length, Mcr and My are cracking and yield moment,
respectively, cr crM EIφ = is the curvature corresponding to Mcr and yφ is the curvature, which
corresponds to My, and is directly determined from moment-curvature analysis of the cross-section.
Since the Eq. (0.7) represents an elastic and plastic part of the rotation the plastic part of the rotation
was determined according to following expression
0, 3
yy p y
M LEI
Θ = Θ − . (0.8)
Rotation at maximum moment and near collapse limit state for columns
Recently a non-parametric empirical approach, called the conditional average estimator (CAE) method
[Grabec and Sachse, 1997], has been implemented for the estimation of the flexural deformation
capacity of reinforced concrete rectangular columns [Peruš, Fajfar, Poljanšek, 2006]. The CAE
method enables relatively simple empirical modeling of different physical phenomena. The method
has been adapted by the authors and has already been used for predicting the capacity of RC walls in
terms of shear strength, ductility, ultimate drift and failure type [Peruš, Fajfar and Grabec, 1994], and
for the modeling of attenuation relationships [Fajfar and Peruš, 1997].
In the CAE method the experimental database is need for prediction of the parameter, which is the
subject of the prediction. According to Peruš, Fajfar and Poljanšek [2006], the PEER Structural
Performance Database complied at the University of Washington, with 156 test specimens, was
selected.
13
Four different parameters were selected in order to determine the rotation at maximum moment mΘ
and the rotation at near collapse limit state ncΘ . These parameters are:
axial load index ( P∗ )
shear span index ( L∗ )
concrete compressive strength (fcm)
confinement index multiplied by confinement effectiveness factor s s sy cmf fαρ αρ∗ = .
The axial load index is defined as the ratio between the axial load P (positive for compression) and the
, where b represents the width of the compression zone, h the depth of rectangular column
in the direction of loading and fcm concrete compressive strength. The shear span index is defined as
Lv/h, where Lv=M/V the length from the location of plastic hinge and the zero moment, and h the depth
of rectangular column in the direction of loading. The confinement effectiveness factor is defined
according to Eurocode 8 [CEN, 2004]
oP bh f= cm
2
1 1 12 2 6
ih h
o o o
bs sb h h
α⎛ ⎞⎛ ⎞⎛ ⎞
= − − −⎜⎜ ⎟⎜ ⎟⎜⎝ ⎠⎝ ⎠⎝ ⎠ob ⎟⎟∑ (0.9)
where bo and ho represents the width and the depth of the confined core (measured between the
centrelines), respectively, sh is the spacing of stirrups and bi is the centreline spacing of the
longitudinal bars (indexed by i) laterally restrained by a stirrup corner or a cross-tie along the
perimeter of the cross-section. The parameter s sx hA bsρ = is the ratio between the transverse steel
parallel to the direction of loading and the area defined as the product of the width of the column and
the spacing of stirrups sh.
The input parameters bl for determination of rotation at maximum moment mΘ and rotation at near
collapse limit state ncΘ , which are [ ]0 0.6P∗ ∈ − , [ ]2 6L∗ ∈ − , [ ]20 120∈ −cmf and
[ ]0 0.14sαρ∗ ∈ − were normalized
,min
,max ,min
l ll
l l
b bb
b b−
=−
(0.10)
where bl,min and bl,max are the selected lower and upper bounds of the input parameters presented above
in the brackets. The rotation Θ, which can be mΘ or ncΘ , is then calculated according to following
equation
1
N
n nn
A=
Θ = Θ∑ (0.11)
14
where N is the number of experiments in the database, nΘ is the rotation, e.g. the rotation at the
maximum moment or the rotation at near collapse limit state, of the n-th experiment from the database
and
1
nn N
ii
aAa
=
=
∑ (0.12)
where
( )
( )2
2 211
1 exp22
Dl nl
n Dl nln nD
b ba
ww wπ =
⎡ ⎤−⎢ ⎥= −⎢ ⎥⋅ ⋅ ⎣ ⎦∑
…. (0.13)
The parameter lb is the l-th input parameter from among P∗ , L∗ , fcm and sαρ∗ , nlb is the l-th
parameter of the n-th experiment from the database and D is the number of the input parameters. The
wnl is the so called smoothing parameter. It determines how fast the influence of data in the sample
space decreases with increasing distance from the point whose coordinates are determined by the input
parameters. The selection of the proper value for wnl is important. According to Peruš, Poljanšek and
Fajfar [2006] the parameter wnl is not constant since the non-constant value of the parameter wnl
improves the prediction of the rotations. It has trapezoidal shape as presented in Figure 2.8.
Figure 2.8. Rule for a non-constant smoothness parameter wnl.
The CAE method employed for the prediction of the rotation at maximum moment and at the limit
state of near collapse has proven to be an efficient method if an appropriate database is available.
More details about the CAE method implemented for the estimation of characteristic rotation for
rectangular columns can be found in Peruš, Poljanšek and Fajfar [2006].
Rotation at maximum moment and at near collapse limit state for beams
The rotation at near collapse limit state for beams was determined according to Eurocode 8-3 [CEN,
2005]
15
( ) ( )( ) (
0.225 0.35100max 0.01, '
0.16 0.3 25 1.25max 0.01,
sys
cm d
ffd v
nc cmel
Lfh
αρρν ωγθ
γ ω⎡ ⎤ ⎛ ⎞= ⋅ ⎢ ⎥ ⎜ ⎟
⎝ ⎠⎣ ⎦) (0.14)
where dγ is the parameter considering the seismic detailing, which can be assumed 0.825 instead of 1
in the members without detailing for seismic resistance. In our example it was assumed 0.85, which is
a suggested value in one of the former Eurocode 8-3. The parameter elγ takes into account the
importance of the structural member. In the case of primary seismic element 1.5elγ = , otherwise 1.0.
The normalized axial force ν is defined like the axial load index P∗ in the previous Section, and is
assumed 0 for beams. Similarly α, ρs, Lv and h have the same meaning as defined in previous Section.
The mechanical reinforcement ratio of the tension ω and the compression reinforcement ω′ are
defined as
sy syst sc
c cm c cm
f fA A,A f A f
ω ω′= ⋅ = ⋅ (0.15)
where Ast and Asc are areas of longitudinal reinforcement in tension and compression, in our case
determined at the maximum moment resulted from moment-curvature analysis. Ac is the gross area of
the cross-section and the fsy and fcm are the yield and the compressive strength of steel and concrete,
respectively. Lastly, the parameter ρd is the steel ratio of diagonal reinforcement in each diagonal
direction.
In order to determine the rotation at maximum moment the ratio between the rotation at total collapse
and the rotation at maximum moment was assumed to be constant for all beams. A value of 3.5 was
adopted. From rotation at near collapse limit state (Eq.(0.14)) and from the assumed ratio between the
rotation at total collapse and the rotation at maximum moment the rotation at maximum moment can
be calculated. Some other researchers [Haselton, 2006] proposed the empirically based formulas for
prediction of the rotation at maximum moment. However the observed dispersion between the
empirical results and results predicted based on empirical formula is very high.
Bond-slip model
Many authors have studied the bond-slip problems by performing experiments or by developing the
model for simulating the bond-slip in the reinforced concrete members [Park and Pauly, 1975,
Filippou et al, 1992, Saatcioglu et al., 1992, Ayoub and Filippou, 1999, Sezen and Moehle, 2003].
These studies have shown that elongation and slip of the tensile reinforcement, especially at the beam-
column interface, could result in significant fixed-end rotations, which are not included in the usual
16
flexural analysis. This phenomenon has been also observed during the pseudo-dynamic test of the
studied structure [Negro et al., 1996] and also considered in this study.
In this study the simple model proposed by Park and Paulay [1975] and used also by Filippou et al.
[1992] was adopted. The model is based on the assumption of uniform bond stress μ along the
development length ld (Figure 2.9). Therefore the stress in the reinforcing bar σs uniformly decreasing
in the region of the length ld (Figure 2.9). It is also assumed that the anchorage length of the bar is
sufficient. It means that the yield stress of the reinforcing bar can be developed (σs=fsy). The force in
the reinforcing bar can be written as
2
4b
s b ddF πσ μ π= ⋅ = ⋅ d l (0.16)
where db is the average bar diameter. The required development length can be simply expressed from
Eq. (0.16)
4
s bd
dl σμ
= . (0.17)
The slip or the elongation of the reinforcing bar, assuming the Hook law σs=Esεs, can be determined
according to following equation
0 0 2
d dl ls s d
ss d s
ls dl l dlE l Eσ σε= = =∫ ∫ . (0.18)
The slip s can be now expressed if the development length lb in Eq. (0.18) is substituted with an Eq.
(0.17)
2 2
8s b
s
dsE
σμ
= . (0.19)
According to Park and Paulay [1975] the uniform bond stress is approximated as 1.35 cmf , where fcm
is the mean concrete strength. The rotation Θs due to the bar slip can be simply calculated as the ratio
between the bar slip and the length between the reinforcement layers (Figure 2.9).
Based on adopted bond-slip model, the additional plastic hinge was modeled at both ends of beam
hinges. The bilinear moment rotation was assumed. The yield moment and the hardening for the
plastic hinge were assumed as calculated from the cross-section analysis of the beam section.
Therefore this model increase the deformation due to the bar slip but does not reduce the overall
strength of structure, since the sufficient anchorage length was assumed.
17
Figure 2.9. Determination of the bar slip.
2.4. Eigen value analysis
The periods and the first three mode shapes are presented in Table 2.7 and 2.8. The first and the
second periods is practical the same as the period, which was measured on the full scale specimen.
The first mode shape is predominantly translational in X direction, second mode is translational in Y
direction and third mode and the third mode is predominantly torsional. The first and the second mode
shape will be used for determination of the horizontal force shape for pushover analysis.
Table 2.7. Periods for the first six modes and the measured period on the full scale specimen.
T1,exp (s) T1 (s) T2 (s) T3 (s) T4 (s) T5 (s) T6 (s)
0.560 0.554 0.550 0.453 0.178 0.176 0.146
2.5. Pushover analysis
The force pattern for pushover analysis is calculated by multiplying mode shape (Table 2.8) and the
storey masses (Section 2.1). The values for force pattern are practically the same for the analysis in X
and Y direction, and amount to 0.28, 0.60, 0.87 and 1, respectively from first to top storey.
The pushover curves are presented in Figure 2.10. The shape of the pushover curves is almost the
same for both directions and senses of loading. The base shear versus weight ratio amounts to about
0.32 for all pushover curves presented in Figure 2.10. The maximum base shear is obtained at the top
displacement of about 30 cm. After this displacement the structure degrades. The 20% reduction of the
maximum base shear is obtained at the top displacement of about 80 and 70 cm, respectively, for
analysis in X direction and Y direction.
18
The drift angles, which correspond to the maximum base shear and to the 80% of the maximum base
shear measured in the degrading part of pushover curves, are presented in Tables 2.9 and 2.10. Quite
uniform distribution of the drifts along the stories is observed. The only exception is the drift at the top
storey, which is smaller in comparison to the drifts obtained in other stories. The maximum drift is
about 0.03 and 0.07, which correspond to the maximum base shear and 80% of the maximum base
shear in the degrading part, respectively.
Table 2.8. Mode shape 1, 2 and 3.
Mode 1
Storey UX UY RZ
1 0.27 0 0.008
2 0.57 0 0.018
3 0.84 0 0.027
4 1.00 0 0.032
Mode 2
Storey UX UY RZ
1 0 0.27 0
2 0 0.57 0
3 0 0.84 0
4 0 1.00 0
Mode 3
Storey UX UY RZ
1 0.19 0 0.271
2 0.41 0 0.578
3 0.59 0 0.843
4 0.68 0 1.000
The yielding of reinforcement, the maximum moment and the rotation at the near collapse limit state
at the hinge level of beams and columns is indicated on the pushover curves (Figures 2.11 and 2.12).
These results can be used for different purpose, for example, to obtain the damage on the structure for
calculated target displacement, or to obtain the αu/α1 ratio, used for determination of the behavior
factor according to Eurocode 8 [CEN, 2005]. In our case the αu/α1 ratio is equal to 1084/751=1.44 and
19
1102/715=1.54 for the analysis in X and Y direction, respectively. Note that the base shear force for
analysis in X and Y direction (positive sign) amounts, respectively, to 1084 kN and 1102 kN, and that
the base shear force, which corresponds to first yielding in beam is 751 kN for the pushover analysis
in X and 715 for pushover analysis in Y direction.
Figure 2.10. Pushover curves for positive and negative X and Y direction.
Table 2.9. Maximum drift angle determined at maximum base shear.
Drift angle at maximum base shear
Storey 1 Storey 2 Storey 3 Storey 4 + X 0.022 0.029 0.027 0.021 - X -0.022 -0.028 -0.027 -0.020 + Y 0.022 0.027 0.025 0.013 - Y -0.022 -0.027 -0.025 -0.014
Table 2.10. Maximum drift angle determined at 80% base shear measured in the degrading part of
pushover curve.
Drift angle at 80% base shear in the degrading part
Storey 1 Storey 2 Storey 3 Storey 4 + X 0.072 0.073 0.062 0.042 - X -0.071 -0.073 -0.065 0.043 + Y 0.071 0.072 0.058 0.026 - Y -0.069 0.071 -0.062 0.026
20
Figure 2.11. The relationship between the damage of plastic hinges and pushover curve for analysis in
X direction. The damage of hinges is indicated by yielding of reinforcement, maximum moment and
rotation at NC limit state for columns and beams.
21
Figure 2.12. The relationship between the damage in plastic hinges and pushover curve for analysis in
Y direction. The damage in hinges is indicated by yielding of reinforcement, maximum moment and
rotation at NC limit state for columns and beams.
22
2.6. Nonlinear dynamic analysis and comparison with pseudo-dynamic test
The nonlinear dynamic analysis was performed for the same ground motion record, which was used in
the pseudo-dynamic test (Figure 2.5). This enables comparison between experimental and calculated
results. Since two pseudo-dynamic tests were performed at ELSA Laboratory also two nonlinear
dynamic analyses were performed in a series. First analysis corresponds to the so called low (L) test,
for which the peak ground acceleration is about 0.12 g. The peak ground acceleration of the 0.45 g,
was used for the second analysis, so called high (H) test. Zero viscous damping was assumed in
pseudo-dynamic test and consequently also in the nonlinear dynamic analysis.
Storey displacements for L and H test are presented in Figures 2.13 and 2.14. Despite very simple
mathematical model very good comparison between measured and calculated storey displacement can
be observed.
The storey shear time histories are presented in Figures 2.15 and 2.16. A close relation between
calculated and measured storey shear forces is observed. Since peak value of the calculated base shear
force is a bit underestimated, if compared with peak value of measured base shear force (Figure 2.16),
it can be concluded that the strength of the structure is slightly underestimated.
2.7. Incremental dynamic analysis
The Incremental Dynamic Analysis (IDA) was developed by Vamvatsikos and Cornell [2002, 2004]
and it is a powerful tool for estimation of seismic demand and capacity for multiple levels of intensity
and can be used for different applications (e.g. [Dhakal et al, 2006], [Fragiadakis et al, 2006]).
However, it requires a large number of inelastic time-history analyses.
The IDA analysis was performed for the same ground motion record as used in the pseudo-dynamic
test (Figure 2.5). The IDA curves, presented in terms of peak ground acceleration versus maximum top
displacement and peak ground acceleration versus maximum storey drift angle, are shown in Figure
2.16. The peak ground acceleration, which causes the dynamic instability of the structure, determined
with the precision of 2%, is about 1.5 g. This is rather high value. However, structure stars degrading
if peak ground acceleration is about 1 g. This can be concluded if capacity diagram is compared with
the IDA curve (Figure 2.16).
23
Figure 2.13. Measured and calculated displacement for the L test.
24
Figure 2.14. Measured and calculated storey displacement for the H test.
25
Figure 2.15. Measured and calculated storey shear forces for the L test.
26
Figure 2.15. Measured and calculated storey shear forces for the H test.
27
Figure 2.16. The IDA curve and the capacity diagram presented for peak ground acceleration versus
maximum top displacement and maximum storey drift angle.
More details regarding the relation between the damage on the structure and the peak ground
acceleration is presented in Figure 2.17. The damage at the hinge level is indicated with yielding of
reinforcement, maximum moment and the rotation at near collapse limit state, which is defined at the
85% of the maximum moment measured in the degrading part of the moment-rotation relationship. It
can be observed that yielding of the reinforcement starts at peak ground acceleration of about 0.2 g,
maximum moment in columns in first storey is reached at peak ground acceleration of 0.6 g and the
near collapse limit state in beam hinges starts at about 1.0 g.
2.8. Determination of the target displacement by N2 method
The N2 method is a simplified nonlinear method for seismic assessment of structures [Fajfar, 2000],
which combines pushover analysis of a multi degree-of-freedom (MDOF) model with the response
spectrum analysis of an equivalent single-degree-of-freedom (SDOF) model. The N2 method has been
implemented in Eurocode 8 (EC8) [CEN, 2005], and extended to infilled frames [Dolšek and Fajfar
2004, 2005] and to probabilistic seismic assessment [Dolšek and Fajfar, 2007].
In this Section the target displacement is calculated according to Eurocode 8 [CEN, 2005] for the
structure subjected to an earthquake in Y direction only. The seismic demand is determined for two
levels of peak ground acceleration, firstly for design peak ground acceleration 0.3 g and also for the
peak ground acceleration 0.45 g, which was used in pseudo-dynamic test (Section 2.2). The spectrum
shape has the shape of the elastic response spectrum according to EC 8. However, the parameters of
28
Figure 2.17. The relationship between the damage in plastic hinges and IDA curve for analysis in Y
direction. The damage in hinges is indicated by yielding of reinforcement, maximum moment and
rotation at NC limit state for columns and beams.
29
The following input data, parameters and/or assumption were used in the determination of target
displacement
the horizontal force shape f is calculated from the most important mode shape Φ for analysis
in Y direction (Section 2.4) and assumed storey masses M used in pseudo-dynamic test
(Section 2.2)
87 0 268 23 386 0 575 49 4
86 0 839 72 283 1 0 83
⎡ ⎤ ⎧ ⎫ ⎧ ⎫⎢ ⎥ ⎪ ⎪ ⎪ ⎪
⎪ ⎪ ⎪ ⎪⎢ ⎥= Φ = =⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎢ ⎥ ⎪ ⎪ ⎪ ⎪⎣ ⎦ ⎩ ⎭ ⎩ ⎭
. .
.f M
. ..
. (0.20)
The defined horizontal force shape pattern was also used in the pushover analysis in Y
direction (Section 2.5). Further on only the results of the pushover analysis in positive Y
direction will be employed.
the mass of equivalent SDOF system is
(0.21) 2 87 0.268 86 0.575 86 0.839 83 227.9 ti im m∗ = Φ = ⋅ + ⋅ + ⋅ + =∑
the transformation to an equivalent SDOF model is made by dividing the base shear and top
displacement of the MDOF model with a transformation factor Γ, which is defined as
*
*
0.268 87 10.575 86 10.839 86 11.0 83 1
1.2790.268 87 0.2680.575 86 0.5750.839 86 0.8391.0 83 1.0
T
T
TT
mL
⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎪ ⎪ ⎢ ⎥ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎩ ⎭ ⎣ ⎦ ⎩ ⎭Γ = = = =
⎧ ⎫ ⎡ ⎤ ⎧ ⎫⎪ ⎪ ⎢ ⎥ ⎪ ⎪⎪ ⎪ ⎪ ⎪⎢ ⎥⎨ ⎬ ⎨ ⎬⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎪ ⎪ ⎪ ⎪⎢ ⎥⎩ ⎭ ⎣ ⎦ ⎩ ⎭
M 1M
ΦΦ Φ
(0.22)
the pushover curve, for analysis in positive Y direction, was idealized by elasto-plastic
relationship. The yield displacement dy was determined in a such way that the areas under the
actual and the idealized force-displacement curves are equal. The displacement dm was
assumed at the maximum force, and at the yield force Fy of the idealized system was assumed
equal to the maximum force. The yield displacement can be then calculated according to EC8
30
244.72 2 0.269 0.093 m1102
my m
y
Ed dF
⎛ ⎞ ⎛ ⎞= − = − =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠ (0.23)
where yield force Fy, displacement at maximum force dm and the deformation energy Eh,
which was calculated up to the displacement dm, were determined from the pushover curve.
The pushover curve and the idealized force-displacement relationship are presented in Figure
2.18. The transformation of the characteristic force-displacement points of the idealized
MDOF system to the characteristic force-displacement points of the SDOF system can be
simply obtained by dividing the forces and displacement of the idealized MDOF system by
factor Γ
the elastic period of idealized system is
* 227.9 0.0932 2 0.87 s
1102y
y
m dT
Fπ π ⋅
= = = (0.24)
the relation between the (spectral) acceleration and the yield force Fy of the MDOF system is
defined as
21102 3.78 m s 0.386 g1.279 227.9
yay
FS
m∗= = = =Γ ⋅ ⋅
. (0.25)
Figure 2.18. The pushover curve for analysis in positive Y direction and the idealized force-
displacement relationship.
31
The target displacement for the SDOF system can be obtained through relationships between the
reduction factor R, the ductility μ , and the period T (the R-μ-T relations). In our case, where we
determine the target displacement for a given seismic loading the unknown parameter is the ductility
demand, which can be according to EC 8 determined as
( )1 1CC
C
T R T TT
R T Tμ
⎧ − + <⎪= ⎨⎪ ≥⎩
(0.26)
where the reduction factor R due to energy dissipation capacity is defined as the ratio of the
acceleration demand Sae in terms of the elastic spectral acceleration for the period T, to the acceleration
capacity Say (i.e. spectral acceleration corresponding to the yield force, Eq.(0.25))
ae
ay
SRS
= (0.27)
The acceleration demand Sae in terms of the elastic spectral acceleration for the period T can be
obtained from the defined seismic loading in terms of the elastic spectral acceleration. For two levels
of peak ground acceleration these values are Sae=0.517 g and Sae=0.775 g (Figure 2.19). The
corresponding reduction factors calculated according to Eqs. (0.27) and (0.25) are R=1.341 and
R=2.011, respectively, for a peak ground acceleration 0.3 and 0.45 g.
Since the period of the idealized system (Eq. (0.24)) exceeds the corner period at the upper limit of the
constant acceleration region of the elastic spectrum (TC) the ductility demand μ is equal to the
reduction factor R. The target top displacement dt is then obtained as the product of the ductility
demand μ and the yield displacement of the idealized system dy and is dt =1.341⋅0.093 m= 12.5 cm for
the peak ground acceleration 0.3 g and dt =2.011⋅0.093 m= 18.7 cm.
The determination of the target displacement can be also graphically presented in acceleration-
displacement (AD) format. In this case all variables are related to the equivalent SDOF system. In
Figure 2.20 the elastic spectrum, inelastic spectrum and capacity diagram are presented for the peak
ground acceleration 0.3 g and 0.45 g. The displacement demand for SDOF system is determined by
the intersection between the capacity diagram and the inelastic spectrum. Note that the inelastic
spectrum (inelastic displacement Sd versus acceleration at yielding of SDOF system Sa) is defined as
32
, aed de a
SS S SR Rμ
= = (0.28)
where Sae is the elastic acceleration spectrum and the elastic displacement spectrum Sde determined
from the elastic acceleration spectrum
2
24de aeTSπ
= S . (0.29)
Figure 2.19. The acceleration spectra used for determination of target displacement.
Figure 2.20. Seismic demand and capacity in AD format for a) peak ground acceleration of 0.3 g and
b) for peak ground acceleration 0.45 g.
33
2.9. Determination of the target displacement by SDOF-IDA approach
The SDOF-IDA approach for determination of the target displacement is based on the same
philosophy as the N2 method. The difference is appears in the determination of the displacement
demand of the SDOF system, which is in this case determined by nonlinear dynamic analysis of the
defined SDOF system and not by employing the R-μ-T relations. This in general enables any shape of
idealized force-displacement relationship and not only ideal elasto-plastic relationship, which was
used in the case of N2 method (Section 2.8). In addition, the displacement demand can be determined
for single ground motion record as well as for a set of ground motion records.
The top displacement is determined for the same ground motion record as used in the dynamic and
IDA analysis (Sections 2.6 and 2.7). Zero damping was assumed in the analysis in order to obtain
comparable results with the results of IDA analysis for the MDOF model (Section 2.7). The pushover
curve was idealized with the four-linear force-displacement relationship as presented in Figure 2.21.
The transformation factor Γ=1.279 as calculated in Section 2.8 was employed also in this case for
transformation of MDOF to SDOF quantities. Similarly the equivalent mass of the SDOF system is
227.9 t (Section 2.8). However, period of the equivalent system T=0.52 s, which in general is not
needed for the prediction of the SDOF-IDA curve, differs from that determined for ideal elasto-plastic
system (T=0.87 s).
Based on the presented quantities of the SDOF system the SDOF-IDA curve was calculated and it is
presented in Figure 2.22. Very good correlation between the top SDOF-IDA and IDA curves is
observed, especially for the peak ground acceleration which is less then 1.2 g.
Figure 2.21. The pushover curve for analysis in positive Y direction and the idealized force-
displacement relationship employed for SDOF-IDA analysis.
34
Figure 2.22. The peak ground acceleration versus top displacement for IDA and SDOF-IDA analysis
in positive Y direction.
2.10. Influence of some modeling uncertainty on the response of structure
In the seismic performance assessment of structures the determination of the seismic hazard and the
selection of the ground motion records probably represent main sources of uncertainty. However, in
addition to the uncertainty related to random nature of earthquakes, which is treated as the aleatory
uncertainty, the epistemic uncertainty can also has an important influence, especially in the case of the
seismic response, which is a subject of many modeling uncertainties [Dolšek, 2007], and since the data
for the seismic response analysis are usually insufficient. Herein only the influence of some modeling
parameters on the seismic response through nonlinear dynamic and pushover analysis is studied. The
results are compared with the results of the original model (Model 0) or with the experimental results.
Three additional models were created (Table 2.11) in order to study the influence of the bar slip in
beams, which was modeled in the original model, the influence of the beam effective width and the
influence of the material strength. In model 1 the bar slip was neglected. In model 2, beams were
modeled as rectangular sections and consequently zero reinforcement from the slab was assumed. In
the last example (model 3), the characteristic strength of concrete and steel was adopted. Therefore the
concrete strength was assumed 25 MPa and the yield strength was assumed 500 MPa. These values are
less than the mean values used in the original model (model 0) and are presented in Tables 2.1 and 2.2.
The fundamental periods and the most important mode shapes are presented in Tables 2.12 and 2.13.
Periods are compared with the measured period. It can be observed that the period for Model 1 is
reduced in comparison with the period of the original model. This is expected results, since no slip is
35
modeled in the beams. On the other hand the period of Model 2 exceeds the period of the original
model for about 14% since the structure is much more flexible if beams are modeled with rectangular
instead of the “T” cross sections. In the last case (Model 3), a slight difference for period is observed if
compared with the period of the original structural model (Model 0). The difference is the
consequence of the model for bond slip, which depends on the material strength.
Table 2.11. Description of the models created for studying the influence of modeling uncertainties.
Model No. Description
0 The original model described in Section 2.3 1 Bar slip was neglected
2 Beams were modeled rectangular, consequently reinforcement from the slab was neglected
3 Characteristic material strength was used (fc=25 MPa, fsy=500 MPa)
Table 2.12. Fundamental periods for all models compared to the measured period.
Experiment Model 0 Model 1 Model 2 Model 3
0.560 0.554 0.495 0.629 0.571
Table 2.13. The mode shape 1 and 2 for all models.
Mode shape 1 (predominant in X direction)
Storey Model 0
Model 1
Model 2
Model 3
1 0.27 0.30 0.24 0.26
2 0.57 0.61 0.54 0.57
3 0.84 0.86 0.81 0.84
4 1 1 1 1
Mode shape 2 (predominant in Y direction)
Storey Model 0
Model 1
Model 2
Model 3
1 0.27 0.30 0.24 0.27
2 0.57 0.60 0.54 0.58
3 0.84 0.86 0.82 0.84
4 1 1 1 1
36
The pushover analysis was performed for all three models to see the difference in strength and
deformation capacity. The force patterns for pushover analyses were calculated as discussed in Section
2.5. Since the difference in modes shapes are not significant also the force patterns are practically the
same for all pushover analyses performed for different models. The pushover curves are presented in
Figure 2.23. Quite big difference can be observed. The deformation capacity in terms of top
displacement is significantly reduced if the slippage of reinforcement is neglected in beams (Model 1).
The highest deformation capacity is observed if beams are modeled with rectangular sections (Model
2). However, in this case the strength is significantly reduced if compared to the strength of the
original model. Similarly also the strength is reduced if the characteristic material properties are used
(Model 3).
Figure 2.23. Pushover curves for different structural models based on analysis in positive X and Y
direction.
37
The storey drifts angles, which were determined at the 80% of maximum base shear measured in the
degrading part of pushover curves, are for different structural models presented in Table 2.14. The
most uniform drifts are observed for Model 2, while for other models the drift angle at the top storey is
significantly reduced in comparison to the drift angles at other stories.
Quite big scatter is also observed if the top displacement and base shear time histories are compared
for different structural models (Figure 2.24). The results are compared also with the experimental
results for the low level (0.12 g) and the high level test (0.45 g) discussed in Section 2.2. The most
deviation from the test results can be observed for models 2 and 3. However in all cases, the maximum
strength is underestimated as already discussed in Section 2.6).
Table 2.14. Maximum drift angles determined at 80% of maximum base shear measured in the
degrading part of pushover curves.
Pushover in positive X direction
Storey 1 Storey 2 Storey 3 Storey 4 Model 0 0.071 0.072 0.058 0.026 Model 1 0.051 0.054 0.049 0.015 Model 2 0.071 0.076 0.074 0.070 Model 3 0.063 0.064 0.059 0.028
Pushover in positive Y direction Storey 1 Storey 2 Storey 3 Storey 4
Model 0 0.071 0.072 0.058 0.026 Model 1 0.051 0.054 0.049 0.015 Model 2 0.071 0.076 0.074 0.070 Model 3 0.063 0.064 0.059 0.028
Table 2.14. Comparison between maximum top displacement and maximum base shear force.
Low level test (0.12 g) High level test (0.45 g) U (cm) U (cm) F (kN) F (kN) Experiment 3.7 21.3 589 1443
Model 0 3.6 24.5 650 1192 Model 1 3.5 24.0 621 1240 Model 2 5.6 19.5 619 1081 Model 3 4.9 20.7 557 1043
38
Figure 2.24. Measured and calculated base shear force and top displacement time histories for
different models and for both low level and high level tests.
39
3. ICONS frame
3.1. Description of the structure
The test structure is a four-storey plane reinforced concrete RC frame. The building had been designed
to reproduce the design practice in European countries about forty to fifty years ago [Carvalho and
Coelho (Eds.), 2001]. However, it may also be typical of buildings built more recently, but without the
application of capacity design principles (especially the strong column - weak beam concept), and
without up-to-date detailing. In such buildings a soft first story effect may appear in bare frame or
even in uniformly infilled frame [Dolšek and Fajfar, 2001].
The elevation, the plan and the typical reinforcement in columns are presented in Figure 3.1. All
beams in the direction of loading are 0.25 m wide and 0.50 m deep. The slab is 0.15 m thick. The
reinforcement for columns in the first and second storey is also presented in Figure 3.1. The
reinforcement is reduced in the top two stories in columns B and C. In column B only three φ12 mm
bars are disposed on each side of the long side of the column. Such configuration results to 6φ12 mm
bars in the cross-section of column B. For column C in the top two stories only 4φ16 mm bars are used
in the corners of cross-section, while the same reinforcement (2φ12 mm bars) is used in the middle of
the long side of the cross-section. The bottom longitudinal reinforcement in beams consists of 2φ12
mm bars. Three φ12 mm bars are used for the top longitudinal reinforcement at the connection to the
column A (Figure 3.1). The top reinforcement in the beams connected to the column B and to the
column D amounts to 2φ12+2φ16 mm bars. Beams, connected from both sides to column C, have the
strongest reinforcement at the top of the beam (2φ12+5φ16 mm bars). The slab reinforcement is φ8/10
cm.
The design base shear coefficient amounted to 0.08. In the design, concrete of quality C16/20 and
smooth steel bars of class Fe B22k (according to Italian standards) were adopted [Carvalho and
Coelho (Eds.), 2001]. The mean strength of concrete amounts to 16 MPa and the mean yield strength
of steel amounts to 343.4 MPa.
3.2. Pseudo-dynamic tests
The studied bare frame was tested in full scale at ELSA Laboratory (Figure 3.2). Two tests, B475 and
B975, with the same ground motion record (Figure 3.3) were performed in a series. In order to obtain
the peak ground acceleration of the acceleration time history, which was used in the pseudo-dynamic
tests, the moderate to high seismic hazard scenario with the return periods of 475 and 975 years were
40
defined. The peak ground acceleration is 0.22 g and 0.29 g, respectively, for the first (B475) and
second (B975) test.
The results of the experiments can be found in ECOEST2-ICONS Report No.2 [Carvalho and Coelho
(Eds.), 2001]. The concentration of the damage was observed in the third storey after the second test
(B975). The maximum drift angle after the second test (B975) in the third storey was 2.41% and the
maximum measured base shear force was 217 kN. Assuming that the mass of the structure is the same
as used in the test (178 t) and that the design base shear coefficient amounted to 0.08 it can be
concluded that the maximum measured base shear force exceeds the design base shear force for about
55%. The ratio between the base shear and the weight of structure amounts 0.124.
Figure 3.1. The view, the plan and the typical reinforcement in columns of the test structure.
Figure 3.2. The tested specimen at ELSA Laboratory.
41
Figure 3.3. The acceleration spectrum and accelerogram for test B475 (ag,max= 0.22g).
3.3. Mathematical modeling
The mathematical model is based on the bilinear moment-rotation relationship assuming the linear
moment degradation after the point which defines the maximum moment. The moment-rotation
envelopes were calculated by using the OS Modeler function CBeamEnvelopeHystereticSlip and
CColumnEnvelopeHystereticSlip.
The yield and the maximum moment in columns were calculated taking into account the axial forces
due to the vertical loading on the frame, which amounts to 9.1 and 8.0 kN/m2 for the bottom three
stories, and for the top storey, respectively. These uniform loads were calculated from defined masses
for the pseudo-dynamic test, which amount 46 t and 40 t for the bottom three stories, and for the top
storey, respectively. The potential reduction in strength (moment) due to insufficient anchorage length
of the reinforcing bars was not considered when determining the moment-rotation envelopes. An
effective slab width of 75 cm and 125 cm [CEN, 2004a] was considered for the short and long beams,
respectively.
The characteristic rotations, which describe the moment-rotation envelope of a plastic hinge used in
the model, were determined according to the procedure described by [Fajfar et al. 2006]. The zero
moment point was assumed to be at the mid-span of the columns and beams. The ultimate rotation Θu
in the columns at the near collapse (NC) limit state, which corresponds to a 20% reduction in the
maximum moment, was estimated by means of the CAE method [Peruš et al. 2006]. The values of
ultimate rotations Θu estimated for weak columns A, B and D (Figure 3.1) (28 to 30 mrad) are
substantially lower than that estimated for the strong column C (41 to 48 mrad). For the beams, the
EC8-3 [CEN 2005] formulas were used for determination of ultimate rotations. Due to the absence of
seismic detailing, the ultimate rotations were multiplied by a factor of 0.85. Low values were adopted
for the confinement effectiveness factor, α=0.5, and for the ratio of the transverse reinforcement,
ρsx=0.002. The parameter γel was assumed equal to 1.0.
42
3.4. Pushover analysis for model H and model T
The pushover analysis was performed in positive and negative X direction. Three different force shape
pattern were used for pushover analysis. They were determined by multiplying the diagonal storey
mass matrix by assumed storey deformation vectors, which were defined to have triangular shape, first
mode shape and uniform shape. The last two are also prescribed for the pushover analysis in Eurocode
8 [CEN, 2004b]. The pushover curves are presented in Figure 3.4.
The shape of the pushover curves are almost the same for both senses of loading. However, the
pushover curves based on the uniform deformation pattern have higher strength and deformation
capacity if compared to other pushover curves. The base shear versus weight ratio amounts to about
0.15, quite higher value as observed in the pseudo-dynamic test (Section 3.2).
More realistic seismic performance assessment can be determined based on the other pushover curves
for which the base shear versus weight ratio is equal and amounts to 0.123 and 0.122, respectively, for
pushover analysis in positive and negative X direction. These values are in very good agreement with
the experimental results (0.124) presented in Section 3.2. The maximum base shear is obtained at the
top displacement of about 8 cm. After this displacement the structure degrades. The 20% reduction of
the maximum base shear is obtained at the top displacement of about 13 cm.
The drift angles, which correspond to the maximum base shear and to the 80% of the maximum base
shear in measured in the degrading part of pushover curves, are presented in Tables 3.1 and 3.2. The
results are presented only for the analysis with assumed triangular and uniform deformation shape. For
the analysis based on the assumption of triangular deformation shape the largest drifts are observed in
the third storey, not only if they correspond to the 80% of the maximum base shear measured in the
degrading part, but also if they corresponds to maximum base shear. The maximum drift, calculated at
the 80% of the maximum base shear in the degrading part of pushover curve, is slightly less than
0.036. This indicates that the soft storey mechanism was formed in the third storey, like also observed
in the pseudo-dynamic test (Section 3.2).
Different collapse mechanism is observed for the pushover analysis performed by assumed uniform
deformation shape pattern. In this case the deformations are concentrated in bottom two stories (Table
3.2), while in the third storey the deformation is only about half of these observed in bottom two
stories. In addition the drift angles determined in the degrading part of pushover curve at 80% strength
are less than the drift angle observed in the case of pushover analysis with assumed triangular force
deformation patter. This indicates that the beams, which in our case have lower deformation capacity
43
in comparison to the columns, are important source of strength degradation. The opposite can was
observed for the pushover analysis based on assumed triangular deformation shape. In this case only
source of deformation capacity are columns of the third storey (Figure 3.5).
Figure 3.4. Pushover curves for model three different force shape pattern. Analysis is performed in
positive and negative X direction.
Table 3.1. Maximum storey drift angles determined at the maximum base shear. Results are presented
for pushover analysis based on assumed triangular and uniform deformation shape pattern
Drift angle at maximum base shear
Storey 1 Storey 2 Storey 3 Storey 4
+ X 0.004 0.006 0.017 0.003 Triangular
– X -0.005 -0.007 -0.017 -0.003
+ X 0.007 0.007 0.005 0.002 Uniform
– X -0.007 -0.007 -0.005 -0.002
Table 3.2. Maximum storey drift angles determined at the 80% of maximum base shear measured in
the degrading part of the pushover curve. Results are presented for pushover analysis based on
assumed triangular and uniform deformation shape pattern.
Drift angle at 80% base shear in the degrading part
Storey 1 Storey 2 Storey 3 Storey 4
+ X 0.004 0.005 0.036 0.002 Triangular
– X -0.004 -0.006 -0.036 -0.002
+ X 0.029 0.028 0.015 0.002 Uniform
– X -0.029 -0.028 -0.015 -0.002
44
Figure 3.5. The relationship between the damage in plastic hinges and pushover curve for assumed
triangular deformation shape. The damage in hinges is indicated by yielding of reinforcement,
maximum moment and rotation at NC limit state for columns and beams.
45
3.5. Nonlinear dynamic analysis and comparison with pseudo-dynamic test
The nonlinear dynamic analysis was performed with intention to simulate pseudo-dynamic test.
Therefore two analyses were performed in a series by subjecting the structure with the acceleration
time history presented in Figure 3.3. The peak ground accelerations amount to 0.22 g and 0.29 g,
respectively, for first (B475) and second (B975) test. The time history results are presented in Figure
from 3.6 to 3.8.
Figure 3.6. The comparison between measured and calculated storey displacement time histories for
the B475 and B975 test (Section 3.2).
46
In Figures 3.6 and 3.7 the time histories for storey displacement and storey drifts are presented.
Despite very simple mathematical model very good relation between measured and calculated results
can be observed. Slightly more difference can be observed for the storey shear time histories.
Figure 3.7. The comparison between measured and calculated storey drift time histories for the B475
and B975 test (Section 3.2).
47
Figure 3.8. The comparison between measured and calculated storey shear force time histories for the
B475 and B975 test (Section 3.2).
48
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