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Engineering Structures 31 (2009) 2865–2872
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Engineering Structures
journal homepage: www.elsevier.com/locate/engstruct
Seismic analysis of reinforced concrete frame-wall systems considering ductilityeffects in accordance to EurocodeJuraj Králik ∗, Juraj Králik JrFaculty of Civil Engineering, Slovak University of Technology in Bratislava, Radlinského 11, Bratislava 813 68, Slovakia
a r t i c l e i n f o
Article history:Received 5 October 2008Received in revised form30 January 2009Accepted 2 July 2009Available online 26 August 2009
Keywords:SeismicReinforced concreteFrame-wallDuctilityEurocode
a b s t r a c t
This paper presents the results of experimental and numerical analysis of the seismic resistance of areinforced concrete coupling system considering the plastic capacity in accordance with the standardrequirements STN ENV 1998 (2005), ENV 1998 (2003), Önorm B4015 (2002) and FEMA 368 (2001). Theplastic capacity of the structure can be established by parameter q in the case of the spectral analysisto determine the seismic response. The experience from dynamic analysis of a hospital structure inaccordance with standard requirements is presented in this paper. Dynamic parameters of the buildingstructure are checked by experiment and the calculationmodel ismodified on the basis of the experiment.The nonlinear analysis of the coupling system was realized in the program CRACK under system ANSYSfor Kupfer’s failure condition and Červenka’s model of the concrete strength reduction.
© 2009 Elsevier Ltd. All rights reserved.
1. Introduction
The new seismic-resistant construction standards [1] and STNP ENV 1998 enable one to consider the seismic load effect in struc-tures with regard to their partial damage, and eventually collapse.With respect to computational and economical complexity
there was a method established, which permits transforming anonlinear dynamic calculation to a linear domain. The process ofthe failure is described by so-called ductility factor. The evaluationof the seismic response based on spectral analysis performs theductility factor to design spectrum, as main feature of the seismicload.The ductility factor is well described in standards, although
nonessential ignorance of physical background could tend to incor-rect results and performance errors [2,3]. Incorrect and unsuitableinterpretation can occur especially when hybrid structural sys-tems, irregular geometry shapes and masses are considered [4–7].Nonlinear behavior of a hybrid system became an aim of
this analysis. The framework of the hospital facility consists of acombination of frames, shear walls and a core wall system (Fig. 1).Nonlinear analyses of walls coupled with frames were realizedby software called CRACK, using Kupfer’s biaxial stress failurecriterion and also the reduction of concrete strength described byČervenka [8]. Program CRACK runs as a subroutine of the softwarepackage ANSYS [9].
∗ Corresponding author. Tel.: +421 252494332.E-mail addresses: [email protected], [email protected] (J. Králik).
0141-0296/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2009.07.029
2. Inelastic design spectrum
If a structure or its elements occurs in the plastic domain,the design spectrum can be reduced by a ductility factor [2].The standardized ultimate strength characterizes elasto-plasticbehavior (Fig. 2).
f y =fyfo=uyuo=1Ry
(1)
where fy anduy are the yield values of the force anddisplacement ofthe structure, fo and uo are peak values of a seismic response forceand deformation at the linear behavior system corresponding to anelasto-plastic system. The value of the parameter fy is evaluated asa function fo through the decrease factor Ry of the ultimate strengthvalue.The ductility factor is defined [2] as a ratio which normalizes
the displacement in relation to the maximum displacement valuein the elastic domain. (For instance there is a case when the firstplastic hinge occurs in the frame.):
µ =umuy
(2)
where the displacement value um defines either a peak or anabsolute value of a seismic displacement induced in the elasto-plastic system.The ductility factor is equal to 1(µ = 1) if fy = fo in the case of
elasto-plastic behavior of the structure. Moreover the following isvalidumuo= µ · f y =
µ
Ry. (3)
2866 J. Králik, J. Králik Jr / Engineering Structures 31 (2009) 2865–2872
Fig. 1. The calculation model of the reinforced concrete structure of the hospital facility.
Table 1The calculation model of reinforced concrete structure of the hospital building.
Period Elastic response spectrum Design response spectrumENV 1998-1 (2003) ENV 1998-1 (2003)
Horizontal spectrum
0 ≤ T ≤ TB Se(T ) = ag .S.[1+ T
TB. (η.2, 5− 1)
]Sd(T ) = ag .S.
[23 +
TTB.(. 2,5q −
23
)]TB ≤ T ≤ TC Se (T ) = ag .S.η.2, 5 Sd (T ) = ag .S. 2,5qTC ≤ T ≤ TD Se (T ) = ag .S.η.2, 5
[TCT
]. Sd (T ) = ag .S. 2,5q
[TCT
]≥ 0, 2.ag .
TD ≤ T Se (T ) = ag .S.η.2, 5[TC .TDT2
]. Sd (T ) = ag .S. 2,5q
[TC TDT2
]≥ 0, 2.ag .
Vertical spectrum
0 ≤ T ≤ TB Sve (T ) = avg .S.[1+ T
TB. (η.3, 0− 1)
]Svd (T ) = avg .S.
[23 +
TTB.(. 2,5q −
23
)]TB ≤ T ≤ TC Sve (T ) = avg .S.η.3, 0 Svd (T ) = avg .S. 2,5qTC ≤ T ≤ TD Sve (T ) = avg .S.η.3, 0
[TCT
]. Svd (T ) = avg .S. 2,5q
[TCT
]≥ 0, 2.avg .
TD ≤ T Sve (T ) = avg .S.η.3, 0[TC .TDT2
]. Svd (T ) = avg .S. 2,5q
[TC TDT2
]≥ 0, 2.avg .
f
f0
uv u0 um
u
fy
Corresponding linear system
Elasto-plastic system
Fig. 2. Elasto-plastic system and its corresponding linear system.
One of the most often used simplified interpretations of f y orRy [2] is the following:
f y =
1(2µ− 1)−1/2
µ−1
Tn < TaTb < Tn < TcTn > Tc
or
Ry =
1(2µ− 1)1/2µ
Tn < TaTb < Tn < TcTn > Tc .
(4)
The factor q described by Eurocode (EC) [1]—standards and alsoby a national standard denoted STN ENV [10] is comparable to thedecrease factor Ry of the ultimate strength value. Factor Ry is usedby the US technical literature and standards [11].
3. Seismic load based on EUROCODE
The termbehavior factor q, presented bynational standards STNENV 1998 [10] and ENV 1998 [1] devoted to seismicity, reflects:failure intensity of the structure depending on its importance, thesubsoil, and ductility and structural regularity classes. Its valueequals 1.0 in the case of linear behavior.Basic relations dedicated to an estimation of the elastic and
design acceleration response spectrum are compiled in Tables 1and2,where Se(T )denotes the ordinate of the design spectrum, T—vibration period of a linear single degree of freedom system, ag—design ground acceleration in the return period of the occurrence,Tb, Tc—limits of the constant spectral acceleration branch, S—soilparameter, q—behavior factor depending on the importance of thestructure,η—damping correction factorwith reference valueη = 1for 5% viscous damping.The recommended values of parameters for spectrum type 1 in
the Table 1 are the following
avg = 0.9ag , TB = 0.05s TC = 0.15s TD = 1. (5)
The q factor established in STN ENV 1998 [10] to 3 is expressedin the following form:
q = qokw > 1.5 (6)
qo—basic value of the behavior factor, depends on the type ofstructure, and the kw—factor reflecting the prevailing failure modein a structural system with walls.Eurocode 8 [1] advises for various structures following ranges of
values for the behavior factor: steel structures, concrete structures,non-bearing structures. In Table 2 there is shown a comparisonof behavior factor values, depending on system of the structure
J. Králik, J. Králik Jr / Engineering Structures 31 (2009) 2865–2872 2867
Table 2Elasto-plastic system and its corresponding linear system.
Ductility factor ‘‘q’’Standard Level Reinforced concrete
Frames multistory regulatory Shear walls Frames & shear walls
STN ENV (2005)DCL 2.5 2.00 2.25DCM 3.75(3.00) 3.00(2.40) 3.38(2.70)DCH 5.00(4.00) 4.00(3.20) 4.50(3.60)
ENV 1998-1-1 (2003)DCL 1 1 1DCM 3.90(3.12) 3.00(2.40) 3.90(3.12)DCH 5.85(4.68) 4.40(352) 5.85(4.68)
FEMA 368 (2001) 3–6 4–5 5.5
ÖNORM B4015 (2002) DCM 1.5 1.5 2.9DCH 3.0 2.5 3.0
CRACK (nonlinear calculation) – – 2.37/2.84
Fig. 3. Stress–strain concrete diagram.
and on criterion ductility. A wide scatter of the ductility factor qvalues contributes to an enhanced demand to verify their accuracyin respect of the type of structural systemandboundary conditions.The ductility magnitude of the structural system depends mostlyon the load level.
4. Nonlinear analysis of the reinforced concrete wall
Nonlinear analysis of the seismic resistance is determined bytheweakestmember of a coupled frame–shearwall hybrid bearingsystem (Fig. 1). It means that: resistance of the entire structuredepends on the critical member’s resistance. The failure in criticalcross-sections can evoke collapse of the entire structure [4,12–15,6,7]. In the case of the hospital facility [16], there is a behaviorfactor determined by the bearing capacity of the horizontal bracingsystem located on the 48th axle, the place of coupling shear wallsand bracing frame.The numericalmodel of the coupled frame–shearwall system is
based on Červenka’s reduction model [8] of the concrete strengthand also on Kupfer’s biaxial stress plasticity criterion in the planeof the principal stresses. A new program was developed calledCRACK [9] containing aforementioned techniques. This programcooperates with the system ANSYS.The constitutive model presented is a further extension of the
smeared crackmodel. The smeared crackmodel, used in this work,results from the assumption, that the field of more micro cracks(not one local failure) brought to the concrete element will becreated. A validity of this assumption is determined by size of finiteelement, hence its characteristic dimension,whereA is the elementarea (versus integrated point area of element).One concrete layer was considered as orthotropic material for
which the direction of a crack is the same as the direction of theprincipal strain.In this model the Kupfer’s bidimensional failure criterion of
concrete is considered. The concrete compressive stress fc , tensileconcrete stress ft and shear modulus G are reduced after thecracking of the concrete.
Fig. 4. Kupfer’s plasticity function.
The stress–strain relation (Fig. 3) is defined following ENV1992-1-1 followingLoading in compression region
εcu < εeq < 0, σ efc = fefc
kη − η2
1+ (k− 2) η, η =
εeq
εc,
(εc =̇ − 0.0022, εcu =̇ − 0.0035).(7)
Softening in compression region
εcm < εeq < εcu, σ efc = fefc
(1−
εeq − εc
εcm − εcu
). (8)
Tension region
εt < εeq < εm, σ efc = ft exp(−2(εeq− εt)/εtm). (9)
In the case of plane stress the strength function in tension and incompression was considered as equivalent values. In the plane ofprincipal stresses can be defined the relation between the one andtwo stress states due to the plasticity function by Kupfer (Fig. 4).Compression–compression
f efc =1+ 3.65a
(1+ a)2fc, a =
σc1
σc2. (10)
Tension–compression
f efc = fcrec, rec =(1+ 5.3278
σc1
fc
), rec ≥ 0.9. (11)
Tension–tension
f eft= ft ret , ret =
A+ (A− 1) BAB
, B = Kx+ A,
x = σc2/fc(12)
ret = 1⇔ x = 0, ret = 0.2⇔ x = 1. (13)
2868 J. Králik, J. Králik Jr / Engineering Structures 31 (2009) 2865–2872
The shear concretemodulusGwasdefined for cracking concreteby Kolmar in the form
G = rg .Go, rg =1c2ln(εu
c1
), 0 < p < 0.02 (14)
c1 = 7+ 333 (p− 0.005) , c2 = 10− 167 (p− 0.005) (15)
where Go is initial shear modulus of concrete, εu is strain in thenormal direction to crack, c1 and c2 are constants dependent onratio of reinforcing, p is the ratio of reinforcing transformed to theplane of the crack.The limit of damage at a point is controlled by the values of the
so-called crushing or total damage function Fu. This is defined inthe principal strain space in terms of the plastic strains at the point,the limit equivalent of plastic strain εup.Function of concrete failure (loss of integrity) can be defined in
dependency to the components of principal stresses in the crackplane of layer ‘‘lth’’ by the function of failure surface F lu. Thus
F lu = Flu(ε
p, εpu, ξ) = 0,
F lu =
√√√√αu [(εp1ξ1
)2+
(εp2
ξ2
)2]− εpu = 0;
23≤ αu ≤ 1
(16)
where
ξ1 = ξ2 = 1 ∀ εp1 < 0, ε
p2 < 0,
ξ1 = ξ2 =
(εpu,tens
εpu,comp
)∀ ε
p1 > 0, ε
p2 > 0,
ξ1 =
(εpu,tens
εpu,comp
); ξ2 = 1 ∀ ε
p1 > 0, ε
p2 < 0,
ξ1 = 1; ξ2 =
(εpu,tens
εpu,comp
); ∀ ε
p1 < 0, ε
p2 > 0.
(17)
For the membrane and bending deformation of the reinforcedconcrete shell structure, we have chosen the SHELL91 layered shellelement, on which we propose a plane stress on every single layer.We propose that the crack in one layer of shell element be
oriented perpendicular to the orientation of principal stresses.Hence, the membrane stress and strain vector depends on thedirection of principal stress and strain in one layer
{εcr} = [Tε] {ε} , {σcr} = [Tσ ] {σ } , (18)
where [Tε], [Tσ ] are transformationmatrices for the principal strainand stress in the direction θ in the layer.
[Tε]
=
cos2 θ sin2 θ sin θ cos θ 0 0sin2 θ cos2 θ − sin θ cos θ 0 0
−2 sin θ cos θ 2 sin θ cos θ cos 2θ 0 00 0 0 cos θ sin θ0 0 0 − sin θ cos θ
(19)[Tσ ]
=
cos2 θ sin2 θ 2 sin θ cos θ 0 0sin2 θ cos2 θ −2 sin θ cos θ 0 0
− sin θ cos θ sin θ cos θ cos 2θ 0 00 0 0 cos θ sin θ0 0 0 − sin θ cos θ
.The strain–stress relationship in the Cartesian coordinates can
be defined depending on the direction of the crack (in the directionof principal stress, resp. strain)
[σcr ] = [Dcr ] {εcr} and therefore
[σ ] = [Tσ ]T [Dcr ] [Tε] {ε} = [D] {ε} .(20)
The stiffnessmatrix of reinforced concrete for the layer ‘‘lth’’ canbe written in the following form[Dl]=[T lc.σ
]T [Dlcr] [T lc.ε]+
n∑j′=1
[T ls]T [Dls] [[T ls]]
(21)
where [Tc.cr ], [Ts] are the transformation matrices for concrete andreinforcement separately.
[Dlcr]=
BlE lx BlµlxyElx 0 0 0 0
BlµlxyElx BlE ly 0 0 0 0
0 0 0 0 0 00 0 0 Glxy 0 0
0 0 0 0Glyzks
0
0 0 0 0 0Glzxks
Bl =
E ly
E ly −(µlxy)2 E lx
(22)
where E lx (or Elx) is Young’s modulus ‘‘lth’’ layer in the direction x
(or y), Glxy, Glxy, G
lxy are shear moduli lth layer in planes xy, yz and
zx; ks is the coefficient of effective shear area
ks = 1+ 0.2A25t2≥ 1.2, (23)
where A is the element area, t is the element thickness.After cracking has occurred, the elasticity modulus and
Poisson’s ratio are reduced to zero in the direction perpendicularto the cracked plane, and a reduced shear modulus is employed.Taking 1 and 2 the two principal directions in the plane ofthe structure, the stress–strain relationship for concrete l-layercracked in the 1 direction, isσ1σ2τ12τ13τ23
l
=
0 0 0 0 00 E 0 0 00 0 Gcr12 0 00 0 0 Gcr13/ks 00 0 0 0 G23/ks
l
ε1ε2γ12γ13γ23
l
(24)
where the shear moduli are reduced by parameter rg1 by Kolmar(14) as follow
Gcr12 = Go.rg1, Gcr13 = Go.rg1, Gcr23 = Go.
When the tensile stress in the 2 direction reaches the value f ′t , asecond cracked plane perpendicular to the first one is assumed toform, and the stress–strain relationship becomes:σ1σ2τ12τ13τ23
l
=
0 0 0 0 00 0 0 0 00 0 Gcr12/2 0 00 0 0 Gcr13/ks 00 0 0 0 Gcr23/ks
l
ε1ε2γ12γ13γ23
l
, (25)
where the shear moduli are reduced by parameter rg1 and rg2 byKolmar (14) as follow
Gcr12 = Go.rg1, Gcr13 = Go.rg1, Gcr23 = Gorg2.
The cracked concrete is anisotropic and these relations mustbe transformed to the reference axes xy. The simplified averagingprocess is more convenient for finite element formulation thanthe singular discrete model. A smeared representation for crackedconcrete implies that cracks are not discrete but distributed acrossa region of finite elements.The smeared crack model, used in this work, results from the
assumption, that the field of more micro cracks (not one local
J. Králik, J. Králik Jr / Engineering Structures 31 (2009) 2865–2872 2869
70
Experiment
60
50
40
30
20
10
0
Forc
e F
[kN
]
Pres
sure
p [
kN/m
^2]
Displacement w [mm] Displacement w [mm]
PresentedExperiment
Presented
Plate D1 Plate D4
Fig. 5. Comparison of experimental and nonlinear numerical analysis of plates.
Fig. 6. The shape of dominant modes of the hospital structural model in X ,Y and Z direction.
failure) brought to the concrete element will be created. A validityof this assumption is determined by size of the finite element,hence its characteristic dimension Lc =
√A, whereA is the element
area (versus integrated point area of element). For expansion ofcracking to be valid the assumption of constant failure energiesGf = const
Gf =∫∞
0σn (w) dw = AG.Lc, w = εw.Lc, (26)
where w is width of failure, σn is the stress in concrete in thenormal direction, AG is the area under the stress–strain diagram ofconcrete in tension. The concrete modulus for the descending lineof the stress strain diagram in tension (crushing) can be describedby Oliver depending on the failure energies in the form
Ec,s =Ec
1− λc, λc =
2Gf EcLc .σ 2max
, (27)
where Ec is initial concrete modulus elasticity, σmax is the maximalstress in concrete tension. From the condition of a real solutionof relation (27) it follows, that the characteristic dimension of theelement must satisfy the following condition
Lc ≤2Gf Ecσ 2max
. (28)
The characteristic dimension of an element is determined bythe size of the failure energy of the element. A theory of concretefailure was implied and applied to the 2D layered shell elementSHELL91.The program CRACK was checked and the results were
compared with the experimental results of Hájek [17,12]. Thereinforced concrete plate D1 was loaded by force F in the centerand plate D4 by pressure p on the area of plate. The comparison ofthe experimental and numerical results are presented in Fig. 5.
5. Experimental modal analysis
The seismic resistance of the hospital structure was checkedin the FEM model which was calibrated on the results of theexperimental modal testing. The numerical modal analysis (Fig. 6)was realized in the program ANSYS. An experimental modalanalysis (Figs. 7–9) of the hospital structure and subsoil wasperformed by Arsenal Research [16]. Stiffness parameters of thesubsoilwere experimentally specified (vP = 1261m/s—velocity oflongitudinal waves vS = 600 m/s—velocity of transverse (shear)waves). The subsoil class was defined as B category of Eurocodescale. Also, then the dynamic shear modulus is following:
Gdyn = v2s ρ = 792 MPa. (29)
The numerical modal analysis was realized for various subsoilrigidities. A comparison of the modal characteristics between thenumerical and experimental analysis is presented in the Table 3.There are presented the principal frequency characteristics andeffective mass ratio of six calculation models in three directions.Five models (Nem1 to 5) are considered without the stiffness ofinterior brick walls. The model ‘‘Nem3m’’ includes the interiorwall stiffness. These models have variable stiffnesses of soil(high, medium and low). The medium soil corresponds to themedian value of soil stiffness obtained from the experimentalmeasurements. The higher (or lower) soil stiffness is equal to theupper (or lower) quantile of the soil stiffness using the normaldistribution function, which is considered for the 95% probabilityof nonexceedance. The dominant frequencies in the horizontaldirection of the X-axis and Y -axis are in the interval fX =0.63–1.88 Hz and fY = 0.69–2.85 Hz and in vertical direction ofthe Z-axis is fZ = 2.00–7.39 Hz (Table 3). The results from theexperimental measurements show, that the dominant frequenciesare following: fX = 1.88 Hz, fY = 3.12 Hz, fZ = 5.56 Hz.The calculation model ‘‘Nem3m’’ with the brick wall stiffness and
2870 J. Králik, J. Králik Jr / Engineering Structures 31 (2009) 2865–2872
Fig. 7. The mode in X direction found experimentally for frequency value equals 1.88 Hz, viscous damping 11.4% [16].
Fig. 8. The mode in Y direction found experimentally for frequency value 3.12 Hz, viscous damping 4.18% [16].
Fig. 9. The mode in Z direction found experimentally for frequency value 5.56 Hz, viscous damping 2.99% [16].
Table 3Critical modes of hospital building.
Model Subsoil Direction X Direction Y Direction ZVert./Horiz. Frequency (Hz) Effect. mass ratio (%) Frequency (Hz) Effect. mass ratio (%) Frequency (Hz) Effect. mass ratio (%)
Nem1 R/R 1.12866 68.095 1.52316 68.533 7.39420 59.346Nem2 R/EH 1.06114 71.799 1.51682 69.231 7.39416 59.551Nem3 EH/EH 0.86421 58.735 0.99064 80.775 5.54769 59.372Nem4 EM/EM 0.69891 79.256 0.78654 91.525 2.77618 93.210Nem5 EL/EL 0.62927 79.610 0.69355 92.000 2.00706 98.464
Model with internal brick wall elementsNem3m EH/EH 1.87976 68.022 2.85207 69.998 5.54867 47.241
Experimental measured critical eigen-frequenciesExperiment 1.88 – 3.12 – 5.56 –
Notes: R–Rigid subsoil, EH–Elastic with the high rigidity, EM– Elastic with the medium rigidity, EL–Elastic with the low rigidity.
high soil stiffness gives us the best results in comparison withthe experimental results (Table 3). However, this case corresponds
to an elastic behavior of the soil and structure. The effect ofthe seismic action depends on the principal frequencies in the
J. Králik, J. Králik Jr / Engineering Structures 31 (2009) 2865–2872 2871
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Acc
el. s
pect
rum
Sa
[m/s
^2]
3.0
2.5
2.0
1.5
1.0
0.5
0.0
Acc
el. s
pect
rum
Sa
[m/s
^2]
Frequency [Hz]
ElasticInelastic q=2.0Inelastic q=4.7
ElasticInelastic q=2.0Inelastic q=4.7
0 3 6 9 12 15 18 21 24 27 30 33Frequency [Hz]
0 3 6 9 12 15 18 21 24 27 30 33
Fig. 10. Horizontal and vertical acceleration spectrum.
Fig. 11. FEM model of frame-wall system in modul 48/A-N (a) Original model (b) Upgraded model.
horizontal and vertical directions (Fig. 10). The peak of thehorizontal (or vertical) acceleration spectrum is in the interval2–7 Hz (or 6–20 Hz). Also, hence the seismic action decreases withthe lower level of the principal frequencies of the structure.In the case of the earthquake even the soil stiffness is reduced
and the stiffness of the interior walls will be degraded too.The shear modulus of the soil is reduced depending on theearthquake’s intensity, i.e. G = (0.7–0.9)Gmax for ag = 0.1 g. Theexperience and experimental results from the investigation of theinteraction of interior brick walls and concrete frame establishesthat the thin interior walls and walls without rigid connectionare damaged during earthquake events and their real stiffness isquickly reduced. The standards (FEMA 368 and Eurocode 8) permitone to calculate the frame-wall interaction effect in the case thatthe connection between them is rigid and the seismic resistance ofthe wall is fulfilled.Also, on the base of numerical analysis of all models the critical
structure element of the hospital building is the frame and wall atthe one side of the building. After the damage of this part of thestructure the building will collapse during the earthquake event.The stiffness of this building part is lower as from the oppositeside. From that reason the building is rotated about vertical axis(Figs. 6–10) and some structure elements between them are moreexposed. Two alternatives of these frame-wall structures (originaland upgraded) was investigated from the point of the capacitycheck in elastic and elasto-plastic state (Fig. 11).
6. Nonlinear analysis of the ductility factor
The nonlinear behavior of the hospital building was inves-tigated depending on the ductility capacity of the reinforcedconcrete shearwall on the building side. Estimated values of equiv-alent horizontal forces were calculated by constraint equations
which conveniently match the reference assumption of horizontaldegree of freedom values along the height of the building. After-wards nonlinear analysis was carried out just for a selected part ofthe building structure (Fig. 11). We considered the capacity of theoriginal and upgraded frame-wall structures.The ductility factor was established from a ratio (2) of the
critical fragment deflection in a linear behavior to the nonlinearvalue (Fig. 12).The ductility factor for the original model is as follows
µ∆|ag=0.64 =∆nonlin
∆lin=0.029060.02246
= 1.29
µ∆|ag=1.25 =∆nonlin
∆lin=0.102370.04320
= 2.37(30)
and for the upgraded structure
µ∆|ag=0.64 =∆nonlin
∆lin=0.027030.02035
= 1.33
µ∆|ag=1.5 =∆nonlin
∆lin=0.128080.04510
= 2.84.(31)
The ductility factor, described by the maximum load equation(2) according to [2,3] is identical to the behavior factor q for periodinterval TB ≤ T ≤ TC . The value of the ductility factor obtained bycalculations is evidently lower than the limit value defined by theEuro code standard. But its value is the most similar to an Austriannational standard called ÖNORMB4015 (2002) [18] recommendedvalue.The seismic load described by the acceleration spectrum in
the Eurocode standard is calculated by equations compiled inTable 1 in accordance to defined behavior factor. A comparison ofacceleration design inelastic spectra on Fig. 10 shows that values
2872 J. Králik, J. Králik Jr / Engineering Structures 31 (2009) 2865–2872
1.6
1.4
1.2
1.0
0.8
0.6
0.4
Des
ign
acce
l. ag
[m
/s^2
]
Maximum displacement [mm]
NonlinearLinear
NonlinearLinear
0 20 40 60 80 100 120 140 160Maximum displacement [mm]
0 20 40 60 80 100 120 140 160
0.2
0.0
1.6
1.4
1.2
1.0
0.8
0.6
0.4
Des
ign
acce
l. ag
[m
/s^2
]
0.2
0.0
a b
Fig. 12. Maximum displacement and design acceleration dependence (a) Original model (b) Upgraded model.
at the range of their peaks (it means the frequency range between2–7 Hz) are significantly reduced, i.e. frequencies which describemain mode shapes of the structural system.
7. Conclusions
The application of behavior factor with respect to the failureof the structure significantly affects the design of structuresin seismic regions [16]. Projects of the structures becomeeconomically efficient. In the case of more variable structures, itmeans that structures with irregular geometry in the horizontalas well as the vertical plane, hybrid structures combined withvarious bearing systems etc. it is necessary to verify the accuracy ofthe behavior factor value. This factor was achieved by a nonlinearcalculation method performed on the weakest element of thestructure [2].Factors of the behavior, described in recent standards, are in
case of some irregular structures unsuitably defined. The perfor-mance of these values would result into incorrect conclusions, asthe article shows. The nonlinear analysis, of the 2D critical sub-structures (wall, frame, core wall,. . .) subjected to the quasi load,presents an acceptably accurate view of its resistance.In this paper was presented the nonlinear analysis of the
concrete structures considered the concrete cracking and crushing,layered approximation of the shell elements with various materialproperties, orthotropic material depending on the direction of therotated cracks and the orientation of reinforced steel, modifiedKupfer’s yield function, degradation of the shear modulus byKolmar depending on the properties of the reinforcement [9].The ductility factor determined from the nonlinear quasi-staticanalysis give us the more accurate results.
Acknowledgements
This survey was solved with support of the Ministry ofEducation in the Slovak Republic within the grant task VEGA1/0849/08.
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