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Contemporary Engineering Sciences, Vol. 9, 2016, no. 26, 1255 - 1272
HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ces.2016.67127
Multicomponent Nonlinear Incremental Dynamic
Analysis of r.c. Spatial Framed Structures
Subjected to Near-Fault Earthquakes
Fabio Mazza, Mirko Mazza and Catia Liguori
Dipartimento di Ingegneria Civile, Università della Calabria 87036, Rende (Cosenza), Italy
Copyright © 2016 Fabio Mazza, Mirko Mazza and Catia Liguori. This article is distributed under
the Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Abstract
Fling-step and forward-directivity effects of near-fault ground motions can produce
long duration intense velocity pulses in the horizontal direction, making unexpected
ductility demand in reinforced concrete (r.c.) spatial framed structures designed in
accordance with current seismic codes. In practice, simplified percentage-rule
methods, in which a coefficient indicates the mutual participation between the two
horizontal ground motions, are generally used to deal with the problem of the
critical incidence angle of bi-directional ground motions. In order to investigate the
orientation of the horizontal components of near-fault earthquakes that yields the
maximum inelastic structural response, this work considers six- and twelve-storey
r.c. framed buildings with symmetric plan are designed for a high-risk seismic
region in line with the provisions of the Italian seismic code. A lumped plasticity
model is used to describe the inelastic behaviour of the r.c. frame members, with
flat surfaces approximating the axial load-biaxial bending moment domain at the
end sections where inelastic deformations are expected. A multicomponent
nonlinear incremental dynamic analysis is carried out with reference to different
incidence angles of bi-directional earthquakes, which are scaled to four intensity
levels to encompass a wide range of structural behaviours. To this end, seven near-
fault ground motions are selected, based on the design hypotheses adopted for the
test structures.
Keywords: R.c. spatial frames, Near-fault ground motions, Axial load-biaxial
bending elastic domain, Multicomponent nonlinear incremental dynamic analysis
1256 Fabio Mazza et al.
1 Introduction
R.c. framed buildings built in a near-fault area and designed according to current
seismic codes can undergo severe, quite unforeseen, structural damage [1-3]
depending on the fault type (e.g. strike-slip or dip-slip) and relative position with
respect to the strike direction of the causative fault [4, 5]. More specifically, the
fling-step is the result of permanent ground displacement that generates one-sided
velocity pulses, in fault-parallel and fault normal directions, respectively, for strike-
slip and dip-slip faults. On the other hand, the forward-directivity produces two-
sided velocity pulses without permanent ground displacement, generally oriented
along the fault-normal direction both for strike-slip and dip-slip faults. Many
studies investigated the critical incidence angle of the two horizontal components
of earthquakes acting on r.c. framed buildings [6-10]. The results indicate that often
the nonlinear dynamic response predicted by seismic codes is not conservative,
deriving from the combination of effects due to the horizontal components applied
separately along each structural axis (e.g. the square root of the sum of the squares,
percentage-rules and sum-of-absolute-values) or considering accelerograms
simultaneously applied along the principal in-plan axes. Further studies suggest that
rotation of earthquakes in a near-fault area to the fault-normal and fault-parallel
directions does not necessarily lead to conservative results in terms of ductility
demand [11, 12]. On the other hand, the evaluation of the ductility demand of
columns proves to be sensitive to the direction of the bending moment axis vector
changing at each step of the loading history [13, 14]. Moreover, current knowledge
about the directionality effects on distribution of the ductility demand along the
building height is limited [15]. To study all these aspects, a multicomponent
incremental dynamic analysis (MIDA), which takes into account randomness on
both record and incidence angle, can be a useful tool [16]. The aim of the present
work is to study, through the use of the MIDA, the effects of directionality of the
horizontal components of near-fault ground motions on the nonlinear response of
medium-to-high rise r.c. framed structures.
2 Test structures
Six- and twelve-storey r.c. symmetric-plan framed residential buildings are
considered for the numerical investigation (Fig. 1). In plan geometric dimensions,
cross-section orientation of the columns and floor slab direction are shown in Fig.
1a, while storey heights and cross-section sizes of beams (deep and flat) and
columns (rectangular and square) are reported in Fig. 1b, with reference to the
perimeter and interior frames. The gravity loads are represented by dead- and live-
loads, equal, respectively, to: 4.8 kN/m2 and 2 kN/m2, for the top floor; 5.7 kN/m2
and 2 kN/m2, for the other floors. The weight of the perimeter masonry-infills,
assumed as non-structural elements regularly distributed in elevation, is taken into
account by considering a gravity load of 2.7 kN/m2. A cylindrical compressive
strength of 25 N/mm2 for the concrete and a yield strength of 450 N/mm2 for the
steel are considered. The fundamental vibration periods of the test structures along
Multicomponent nonlinear incremental dynamic analysis 1257
the in-plan principal directions are: T1X=0.576s and T1Y=0.698s, for six storeys;
T1X=0.993s and T1Y=1.249s, for twelve storeys. Further details on their main
dynamic properties can be found in previous work [10].
(a) Plan. (b) Elevations.
Fig. 1. R.c. test structures (units in cm).
Six (i.e. F6) and twelve (i.e. F12) framed structures are designed in line with the
Italian seismic code (NTC08, [17]), considering the following design assumptions
for the horizontal seismic loads: high-risk seismic region; medium subsoil (class B,
subsoil parameter: SS=1.13); flat terrain (class T1, topographic parameter: ST=1).
Criteria imposed by NTC08 for the variation of lateral stiffness and shear strength
are not satisfied at all storeys, so that F6 and F12 structures have to be classified as
1258 Fabio Mazza et al.
irregular in elevation [18]. As a consequence, the ductility class B is considered,
assuming qH=3.12 as behaviour factor for the horizontal seismic loads. The design
of the frame members complies with the serviceability (i.e. damage) and ultimate
(i.e. life-safety) limit states, assuming design earthquakes with 50% and 10%
probability of being exceeded over the reference life of the structure VR=50 years.
Main data of the corresponding horizontal (elastic) design spectra are reported in
Table 1: i.e. return period (TR) of the earthquakes, corresponding to VR; peak ground
acceleration (PGAH); amplification factor defining the maximum spectral
acceleration on rock-site (FH); upper limit of the vibration period of the constant
spectral acceleration branch in the horizontal direction (T*C). Capacity design rules
regarding the beam-column moment ratio, shear forces and local ductility
requirements of r.c. frame members, in terms of minimum conditions for the
longitudinal bars and maximum value of the normalized axial force in the columns,
are also satisfied [12].
Table 1. Main data of the horizontal (elastic) design spectra [17].
Limit state TR (years) PGAH (g) FH T*C (s)
Damage 50 0.108 2.28 0.301
Life-safety 475 0.312 2.44 0.370
3 Method of analysis and near-fault earthquakes
The frame member of the three-dimensional structure shown in Fig. 2a is
described by the coordinates of the end sections (i and j), in the space defined by
the global Cartesian system (X, Y, Z). In the clockwise local system (x, y, z),
corresponding to the principal axes of the frame member, EA is the axial stiffness,
EIy and EIz represent the flexural stiffness around the local axes y and z and GJt is
the torsional stiffness, while the shear deformations are neglected. Uniformly
distributed mass () and vertical load (pz) are considered along the length of the
frame member. The inelastic behaviour of the r.c. frame member is idealized by
means of a lumped plasticity model constituted of two parallel elements (Fig. 2b),
one elastic-perfectly plastic (1) and the other linearly elastic (2), assuming a bilinear
moment curvature law (Mr-r) in the radial direction depending on the axial force
(N). The elastic component of the (Mr -r) law is characterized by the flexural
stiffness pEIr, p being the hardening ratio, while the elastic-perfectly plastic
component exhibits inelastic deformations, lumped at the end sections. Finally,
torsional strains are assumed to be elastic (i.e. Mt=MtE).
For each ground motion, two horizontal accelerograms are first rotated along the
principal directions (X and Y) of the building plan (Fig. 2a)
g ,X g ,H1 g ,H1u ( t ) u ( t )cos u ( t )sin (1a)
Multicomponent nonlinear incremental dynamic analysis 1259
g ,Y g ,H1 g ,H 2u ( t ) u ( t )sin u ( t )cos (1b)
being the incidence angle of the earthquake in degrees counter-clockwise from
the X axis (Fig. 1a). Then, the nonlinear seismic analysis of the framed structure is
carried out using a step-by-step procedure. At each step, the elastic-plastic solution
is derived by the Haar-Kàrmàn principle, without satisfying nodal equilibrium
conditions [13, 14].
(a) Structural discretization. (b) Lumped plasticity model.
Fig. 2. Modelling of r.c. three-dimensional framed structure.
In order to comply with the dynamic equilibrium of the spatial framed structure,
an implicit two-parameter integration scheme and an initial stress-like iterative
procedure are considered [12]. More specifically, the following dynamic
equilibrium equation has to be satisfied
X g ,X Y g ,Yu( t ) u( t ) [ ( t )] u ( t ) u ( t ) M C f u p M i i (2)
which represents a nonlinear implicit system in the unknown velocity vector u ,
where: f is the structural reaction vector; p is the external load vector; iX and iY are
the vectors of the influence coefficients along the horizontal global axes; ≤is
the multiplier of the ground accelerations (see Fig. 2a) in the multicomponent incre
1260 Fabio Mazza et al.
mental dynamic analysis (MIDA). According to the Rayleigh hypothesis, the
damping matrix C is assumed to be a linear combination of the consistent mass
matrix (M) and the elastic stiffness matrix (K), assuming a suitable damping ratio
associated with two control frequencies (or modes).
At each critical end-section of a frame member, the elastic-plastic solution
referring to the k-th flat surface describing the axial load (N) and biaxial bending
moment (MyMz) elastic domain
T
EP,k EP,k EPy,k EPz,k= N ,M ,M (3)
is the one at a minimum distance from the elastic solution (E), in terms of
complementary energy
D
1T -1
c EP EP E EP E
0
L= d =min.
2 (4)
where(=x/L) is a nondimensional abscissa, L the length of the beam element and
D the elastic matrix of a beam element. Moreover, the plastic admissibility
conditions have to be satisfied (Fig. 3)
for k EP fsg 0 k=1..n (5)
In particular, the elastic-plastic solution, represented by point P (Fig. 3), is
obtained as projection of the elastic solution, represented by point E, on the active
flat surface (Fig. 3a) or along the active line (Fig. 3b) or at the active corner (Fig.
3c) resulting from the intersection of surfaces. In the proposed model, the axial
load-biaxial bending moment elastic domain is discretized by 26 flat surfaces,
including (Fig. 3d): 6 surfaces normal to the principal axes; 12 surfaces normal to
the bisections of the principal planes; 8 surfaces normal to the bisections of the
octants. Further details of the numerical procedure to evaluate the elastic domain of
r.c. cross sections can be found in previous work [19].
(a) (b)
(c) (d)
Fig. 3. Elastic-plastic solution of r.c. frame member.
Multicomponent nonlinear incremental dynamic analysis 1261
Long-duration horizontal pulses due to fling-step (FS) forward-directivity (FD)
widely recognized as the main effects of near-fault ground motions. Structural
damage due to resonance, when the predominant vibration periods of both site and
structure are close, is expected in the case of r.c. framed structures subjected to
near-fault earthquakes [20-23]. In order to study their critical incidence angle in the
nonlinear analysis of medium-to-high rise r.c. framed structures, seven near-fault
earthquakes are selected in the Pacific Earthquake Engineering Research center
database [24] on the basis of the design hypotheses adopted for the test structures
(i.e. high-risk seismic region and medium subsoil class). In Table 2 the main data
of the selected near-fault ground motions are shown [25]: i.e. date; recording station;
pulse-type; magnitude (Mw); closest distance to fault rupture (); horizontal peak
ground accelerations (i.e. PGAH1 and PGAH2). The corresponding elastic response
spectra of acceleration for the H1 and H2 components are plotted in Figs. 4a and
4b, respectively, assuming an equivalent viscous damping ratio (H) equal to 5%.
Table 2: Main data of the selected near-fault earthquakes.
Earthquake Station Pulse-type Mw [km] PGAH1 PGAH2
Parkfield, 1966 Cholame 2WA FD 6.0 6.6 0.26g 0.48g
Morgan Hill, 1984 Gilroy STA #6 FD 6.1 6.1 0.22g 0.29g
Loma Prieta, 1989 LGPC FD 7.0 3.5 0.56g 0.60g
Erzincan, 1992 Erzincan FD 6.7 2.0 0.82g 0.60g
Cape Mend., 1992 Petrolia FD 7.1 15.9 0.56g 0.60g
Kobe, 1995 KJMA FD 6.9 0.6 0.82g 0.60g
Chi-Chi, 1999 TCU084 FS 7.6 11.4 0.42g 0.98g
(a) H1 Horizontal components. (b) H2 Horizontal components.
Fig 4. Acceleration (elastic) response spectra.
1262 Fabio Mazza et al.
4 Numerical results
To evaluate the influence of the direction of near-fault ground motion on the
nonlinear seismic response of r.c. framed buildings, an extensive numerical
investigation is carried out by multicomponent incremental dynamic analysis
(MIDA) of the six- (i.e. F6) and twelve-storey (i.e. F12) test structures (see Section
1) subjected to seven pairs of horizontal accelerograms selected in near-fault areas.
Given the lack of knowledge of the direction from which ground motions occur, the
maximum ductility demand of r.c. frame members is evaluated with reference to
different incidence angles. In particular, the direction of the seismic loads is defined
by horizontal (orthogonal) axes rotated of the angle with respect to the in-plan X
and Y principal directions (see Equations 1(a) and 1(b)), while, to encompass a wide
range of structural behaviours, each pair of accelerograms is scaled to different
levels of intensity through the use of the multiplier (see Equation (2)). A computer
code for the nonlinear MIDA of r.c. three-dimensional framed structures is
developed in C++, in line with the lumped plasticity model, based on the Haar-
Kàrmàn principle, described in the Section 2. This includes a piecewise
linearization of the axial load-biaxial bending moment elastic domain of the r.c.
cross-sections of the frame members, with hardening ratio p=5%. In the Rayleigh
hypothesis, the damping matrix is evaluated by assuming a viscous damping ratio
equal to 5%, with reference to the two vibration periods with prevailing components
in the X (i.e. T1X) and Y (T1Y) directions.
Plastic conditions are checked at the potential critical end sections of beams and
columns, where information on the structural damage is obtained by referring to the
curvature ductility demand. In particular, top and bottom maximum values of
ductility demand are evaluated for the beams; while for the columns the ductility
demand is evaluated with reference to the radial direction, being sensitive to the
direction of the bending moment axis vector, which changes at each step of the
loading history
2 2
Ey Py Ez Pzmax,rmax,r
Er Pr V r
χ + Δχ + χ + Δχχμ = =
χ M N EI
(6)
where max,r and E,r represent maximum and yielding curvatures, respectively, in
the radial direction. Plastic curvatures increments at each step of the MIDA analysis
(i.e. Py and Pz) are accumulated and added to the yielding curvatures at the
current step (i.e. Ey and Ez). Finally, the plastic moment MPr is calculated by
considering the axial force due to the gravity loads only (NV) and referring to the
radial direction identified by max,r.
Firstly, domains of the maximum global ductility demand for r.c. frame members
of the F6 and F12 test structures subjected to the selected near-fault ground motions
are reported in the Figs. 5-8. More specifically, as shown in Fig. 1a, the horizontal
components of acceleration of each earthquake are applied along different
incidence angles (i.e. increasing in the range 0-360°, with a constant step of 30°)
Multicomponent nonlinear incremental dynamic analysis 1263
and four levels of earthquake intensity are considered for each angle (i.e. =0.25-
1.0, with a constant step of 0.25).
Fig. 5. Maximum global ductility demand in beams of the F6 structure.
It is worth noting that the nonlinear dynamic analyses are terminated once the
maximum value imposed on the ultimate ductility demand of the frame members is
exceeded. As expected, the “strong-column weak-beam” mechanism provided by
NTC08 [17] is achieved for each intensity level and all directions of the seismic
loads, with ductility demand for deep beams (Figs. 5 and 7) higher than that
observed for rectangular columns (Figs. 6 and 8). Moreover, note that the incidence
angle () that produces the maximum value of ductility demand in the F6 and F12
structures is strongly affected by the characteristics of the near-fault ground motions
and, for each of them, by the intensity level () considered in the MIDA. On the other hand, in both structures, the maximum seismic demand direction for the beams
1264 Fabio Mazza et al.
does not necessarily coincide with that evaluated for the columns. Further results,
which are omitted for the sake of brevity, confirm that the lower values of the
ductility demand observed for the flat beams and the square columns are obtained
for different critical incidence angles.
Fig. 6. Maximum global ductility demand in columns of the F6 structure.
Afterwards, graphs similar to the previous ones are plotted in Figs. 9 (i.e. F6
structure) and 10 (i.e. F12 structure), where the mean of the maximum global
ductility demand separately obtained for the selected near-fault ground motions is
evaluated for each incidence angle and intensity level . As expected, structural
behaviour proves to be more regular in terms of mean values, with similar values
of ductility demand for different in-plan directions of the seismic loads, especially
in beams (Fig. 9a) and columns (Fig. 9b) of the F6 structure.
Multicomponent nonlinear incremental dynamic analysis 1265
Fig. 7. Maximum global ductility demand in beams of the F12 structure.
Only slight irregularities can be observed in the F12 structure, considering some
levels of earthquake intensity for beams (i.e. =0.25, Fig. 10a) and columns (i.e.
=0.25 and =0.50, Fig. 10b). The above considerations seem to indicate therefore
that, for medium-to-high rise r.c. framed symmetric plan structures subjected to
near-fault ground motions, a limited influence of the incidence angle to be expected
on the average global damage level of the building.
1266 Fabio Mazza et al.
Fig. 8. Maximum global ductility demand in columns of the F12 structure.
Finally, to investigate the local damage of the r.c. frame members, in Figs. 11
and 12 curves obtained for the test structures F6 and F12, respectively, subjected to
near-fault earthquakes acting along the principal in-plan directions (i.e. solid lines,
MIDAX-Y) and the critical incidence angle (i.e. dashed lines, MIDA), are compared.
More precisely, the maximum of the mean values of the ductility demand obtained
for deep beams (Figs. 11a and 12a) and rectangular columns (Figs. 11b and 12b) is
plotted along the frame height. It should be noted that, with accelerograms applied
simultaneously along the structural axes of the building, the MIDAX-Y is performed
for two incidence angles of the seismic loads (i.e. =0° and =90°) and mean of
the corresponding maximum values is considered.
Multicomponent nonlinear incremental dynamic analysis 1267
(a) Beams. (b) Columns.
Fig. 9. Mean global ductility demand of the F6 structure.
(a) Beams. (b) Columns.
Fig. 10. Mean global ductility demand of the F12 structure.
On the other hand, the MIDA takes into consideration a critical incidence angle
(cr), whose value is different for beams and columns and each intensity level (
1268 Fabio Mazza et al.
and is evaluated with reference to the mean demand domains of global ductility
previously defined (see Figs. 9 and 10). As can be observed, the directionality
effects of near-fault earthquakes do not significantly affect the vertical distribution
of mean ductility demand, with values obtained with the MIDA slightly more
conservative than those evaluated with the MIDAX-Y, for both F6 and F12 structures
and all the examined values of .
(a) Beams. (b) Columns.
Fig. 11. Mean local ductility demand of the F6 structure.
(a) Beams. (b) Columns.
Fig. 12. Mean local ductility demand of the F12 structure.
Multicomponent nonlinear incremental dynamic analysis 1269
5 Conclusions
The nonlinear seismic response of six- and twelve-storey r.c. framed structures,
representative of medium-to-high rise symmetric plan buildings designed according
to NTC08, has been studied in order to evaluate the effects of different incidence
angles of the horizontal components of near-fault ground motions. To this end, a
computer code for the multicomponent incremental dynamic analysis (MIDA) of
spatial framed structures has been implemented in C++. A lumped plasticity model
is considered to describe the inelastic behaviour of r.c. beams and columns,
including a 26-flat surface approximation of the axial load–biaxial bending moment
elastic domain, at the end sections where inelastic deformations are expected. The
following conclusions can be drawn from the results.
Demand domains in terms of maximum global ductility show that the seismic
performance of the F6 and F12 structures is strongly affected by the characteristics
of near-fault ground motions and, for each of them, by the intensity level considered
in the MIDA. Moreover, the maximum seismic demand directions corresponding
to flat and deep beams do not necessarily coincide and are generally different from
those obtained for square and rectangular columns. On the other hand, demand
domains in terms of mean global ductility prove to be more regular, especially in
the beams and columns of the F6 structure, with similar ductility demand values for
different in-plan directions of the seismic loads and levels of intensity. Moreover,
the vertical distribution of mean ductility demand is not significantly affected from
the directionality effects of near-fault earthquakes, with mean values obtained with
the MIDA slightly more conservative than those evaluated with the MIDAX-Y.
Before drawing any firm conclusions, however, further studies are needed to extend
the analysis to other structures and a large number of recorded near-fault ground
motions.
Acknowledgements. The present work was financed by Re.L.U.I.S. (Italian
network of university laboratories of earthquake engineering), according to
“Convenzione D.P.C.–Re.L.U.I.S. 2014–2016, WPI, Isolation and Dissipation”.
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Received: August 11, 2016; Published: September 19, 2016