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