three-dimensional reinforced concrete structures subjected ...8gracm.mie.uth.gr/papers/session...

8th GRACM International Congress on Computational Mechanics

Volos, 12 July – 15 July 2015

THREE-DIMENSIONAL REINFORCED CONCRETE STRUCTURES SUBJECTED

TO MAINSHOCK-AFTERSHOCK EARTHQUAKE SEQUENCES

Maria P. Hatzivassiliou1, George D. Hatzigeorgiou

2

1Freelance consulting engineer

Thessaloniki, GR-56533, Greece

e-mail: [email protected]

2School of Science and Technology

Hellenic Open University

Patras, GR-26335, Greece

e-mail: [email protected] ; web page: www.eap.gr/view.php?artid=3755

Keywords: Inelastic analysis, three-dimensional structures, seismic sequences.

Abstract. Seismic sequences have in many cases a great influence on the response of the structures and lead to

increased demands in comparison with the single ground motions. Most of the pertinent studies are limited

either to single-degree-of-freedom systems or to planar multi-degree-of freedom systems. The inelastic response

of three-dimensional reinforced concrete (RC) buildings subjected to multiple earthquakes has not yet examined in

depth. In this paper, three-storey and five-storey buildings, regular and irregular along their height, are

investigated herein under real strong repeated earthquakes where all their three components are taken into

account. Parameters such as maximum displacements, maximum residual displacements, maximum interstorey

drift ratio, maximum residual interstorey drift ratio, damage indices and ductility demands, are investigated.

Finally, the building structures under consideration are analyzed for different siting configurations to

investigate the effect of earthquake direction incident. It is concluded that the multiplicity of earthquakes

strongly affect the inelastic response of structures and cause damage accumulation of structural and non-

structural damage as well as the increment of deformation demands.

1 INTRODUCTION

Generally, main-shocks of earthquakes are followed by series of aftershocks of medium or large intensity.

When structures are subjected to multiple earthquakes, in many cases there is a small time period between main-

shock and aftershocks where any rehabilitation action appears to be impractical and impossible. As a result, there

is a significant damage accumulation during the seismic sequences. Despite that fact, the modern seismic design

codes have not yet considered adequately this phenomenon and they typically focus on the single and rare

‘design earthquake’.

To evaluate the additional damage caused by the multiplicity of earthquakes many researchers focused on

single-degree of freedom (SDOF) systems and on multi-degree of freedom (MDOF) systems. All these studies

have been limited to two-dimensional/planar structures and there is not research work that has examined the

effects of repeated earthquakes on three-dimensional reinforced concrete (RC) structures.

This research study investigates the phenomenon of seismic sequences and their effects on three-dimensional

RC structures. For the inelastic analysis Ruaumoko structural analysis program [1] is used. The aforementioned

seismic sequences have been recorded by the same station and in a short period of time, up to three days. The

study focuses on structural damage, displacements, permanent displacements and interstorey drift ratios. In order

to examine the effect of earthquake direction incident, the building structures under consideration have been

analyzed for different siting configurations.

2 DESCRIPTION OF STRUCTURES AND MODELING ASSUMPTIONS

2.1 Description of structures

In this study, four three-dimensional RC structures are investigated. Two of them have 3 storeys, where the

first one is regular and the second one is irregular along its height. The other two have 5 storeys and they are also

regular and irregular along their height. The 3-storey irregular building has a setback on the third floor and the 5-

Maria P. Hatzivassiliou, George D. Hatzigeorgiou

storey irregular building has setbacks on its fourth and fifth floor. The examined 3- and 5-storey buildings have 2

equal bays in each direction (x and y) with total length equal to 10.40 m. Typical floor-to-floor height is equal to

3.0 m. The columns have square sections and their sectional dimension are 40x40cm, 50x50cm and 60x60cm.

The dimensions (section width/height) of beams for the 3- and 5-storey regular buildings are 25x50cm and of

beams for the 3- and 5-storey irregular buildings are 25x60cm, 25x55cm and 25x50cm. The 3-D models of the

structures are shown in Figs 1-4, which also depict the numbering of nodes and elements.

Figure 1. 3-storey regular building. Figure 2. 3-storey irregular building.

Figure 3. 5-storey regular building. Figure 4. 5-storey irregular building.

Reinforced concrete with concrete grade C20/25 and longitudinal and transverse reinforcements with grade

B500C are used. The structures are analyzed and designed for earthquake loads (E) with peak ground

acceleration PGA=0.24g and soil class B, with static analysis StereoSTATIKA [2], according to Eurocodes EC2

[3] and EC8 [4]. The structural analysis and design takes into account dead loads, G, live loads, Q, earthquake

loads, E, as well as any probable loading combination according to the aforementioned structural codes. The

combination coefficient of live load Q is taken equal to 0.3. The design of columns satisfies the following

condition of EC8 [4] at all joints of beams with columns, i.e.,

Rc RbM 1.3 M (1)

where RcM and RbM are the sum of the resistance moments of columns and beams at every beam-column

joint, respectively.

The behavior factor is equal to q=3.6 for buildings with regular elevations and q=0.8*3.6=2.88 for irregular

buildings, since according to EC8 [4], for structures which are not regular in elevation, the value of behavior

factor should be reduced by 20%.

2.2 Nonlinear structural behavior

To simulate the cyclic behavior of RC members in plastic hinge under load reversals, the modified Takeda

hysteresis rule is adopted (see Fig. 5), assuming the following parameters: unloading parameter =0.25 for

beams and =0.50 for columns, reloading parameter =0 both for columns and for beams, and post-yield

X

Y

Z


stiffness ratio r=0.01 for all structural members. The inelasticity is modeled with flexural plastic hinges located

at the member ends and the plastic hinge length, pll , is assumed for each member, equal to the half of its

section’s height H (or width B in perpendicular direction), i.e., pll H / 2 (or pll B / 2 ) [5].

Figure 5. The modified Takeda hysteresis model.

2.3 Other modeling assumptions

In order to perform nonlinear dynamic analysis each structure is described by a three dimensional model in

Ruaumoko [1]. The soil-structure interaction phenomenon is not taken into account, considering fixed base

conditions. The Rayleigh approach for damping is used [6]. Beams are assumed not to be able to deform along

their axis, in order to simulate the diaphragm action of slabs. Furthermore, reduced values of member moments

of inertia, effI , were considered in the design to account for the expected cracking of a concrete section, i.e., for

beams eff gI 0.35I and for columns eff gI 0.60I [5], where gI is the moment of inertia of the cross section. For

the section modeling, the XTRACT program [7] is used.

2.4 The effect of earthquake direction incident

For every irregular building, four different angles between the principal axes of the structures and the

components of the earthquake excitations are investigated: a) 00 , b) 090 , c) 0180 , d) 0270 . The 4 different

configurations under consideration for the 3-storey irregular building are shown in Fig. 6.

Figure 6. Different siting configurations for the 3-storey irregular structure.

The regular buildings, due to their symmetry in XY plane, are analyzed without the aforementioned siting

rotation.

3 SEISMIC INPUT

The corresponding records used in this research have been downloaded from the strong motion database of

the Pacific Earthquake Engineering Research (PEER) Center [8] and consists of five real seismic sequences, which are namely: Mammoth Lakes, Chalfant Valley, Coalinga, Imperial Valley and Whittier Narrows. The

complete list of these seismic events is shown in Table 1.

Each earthquake consists of the two horizontal records and the vertical one. The horizontal component with

the higher peak ground acceleration (PGA) is set as Component (1), the other one as Component (2) and the

vertical axis corresponds to Component (3). Table 1 also provides with the code names of ground motions

examined herein, e.g., CH1, CH2 and CHT correspond to the first and second individual Chalfant Valley seismic

events as well as their seismic sequence, respectively (i.e., CHT=CH1+CH2, etc.). The aforementioned seismic

events have been appropriately scaled [9] to give, for the fundamental period of each structure, identical spectral

acceleration (for Component (1)) with the design spectrum of EC8 [4]. Thus, all these ground motions are

multiplied by appropriate factors ranged between 0.454 ~ 2.939 (i.e., about 0.5~3.0), which are shown in Table


2.

These values can be characterized as rational and acceptable according to the basic principles of Engineering

Seismology, [10]. Every sequential ground motion records from the PEER database [8] becomes a single ground

motion record (serial array) where between two consecutive seismic events a time gap is applied, which is equal

to 100s. This gap has zero acceleration ordinates and is absolutely enough to cease the moving of any structure

due to damping.

No Seismic sequence Station Date (time) Code name Recorded PGA(g)

1 Mammoth Lakes 54099 Convict

Creek

1980/05/25 (16:34) MA1 0.442

1980/05/25 (16:49) MA2 0.178

1980/05/25 (19:44) MA3 0.219

1980/05/25 (20:35) MA4 0.432

1980/05/27 (14:51) MA5 0.316

2 Chalfant Valley 54428 Zack

Brothers Ranch

1986/07/20 (14:29) CH1 0.285

1986/07/21 (14:42) CH2 0.447

3 Coalinga 46T04 CHP 1983/07/22 (02:39) CO1 0.605

1983/07/25 (22:31) CO2 0.733

4 Imperial Valley 5055 Holtville

P.O.

1979/10/15 (23:16) IM1 0.253

1979/10/15 (23:19) IM2 0.211

5 Whittier Narrows 24401 San

Marino

1987/10/01 (14:42) WI1 0.204

1987/10/04 (10:59) WI2 0.212

Table 1: Seismic input data – Real seismic sequences.

Structure Mammoth Lakes Chalfant Valley Coalinga Imperial Valley Whittier Narrows

3-storey reg. 1.369 0.521 0.592 1.590 2.939

3-storey irreg. 1.189 0.641 0.454 0.908 1.564

5-storey reg. 1.816 0.744 1.051 1.262 1.230

5-storey irreg. 1.212 0.572 0.886 1.293 1.852

Table 2: Scaling factors.

Every structure is analyzed both for single earthquakes and seismic sequences. Furthermore, every irregular

structure has been analyzed for all the examined siting configurations that have been presented in Section 2.4.

4 SELECTED RESULTS

This section focuses on damage indices, horizontal displacements, interstorey drift ratios, residual horizontal

displacements, residual interstorey drift ratios and ductility demands of the examined RC three-dimensional

buildings under seismic sequences. The time-history of horizontal displacements for the top storey of the

structures is also investigated.

4.1 Structural damage

The Park-Ang model [11] is the best known and most widely used damage index (DI). This DI takes into

account both the maximum deformation and the hysteretic energy of structural members and can be defined as

m h

u y u

μ bEDI

μ M φ (2)

where mμ represents the maximum ductility of the element, μu is its ultimate ductility, b is a model constant

parameter set equal to 0.05 [5] to control strength deterioration, hE is the hysteretic energy absorbed by the

element during the earthquake, and yM is the yield moment of the element. The ductility is defined in terms of

curvature. Figure 7 depicts the DI for single seismic events and seismic sequences for characteristic structural

members. It is obvious that seismic sequences lead to an increment of DI in comparison with single seismic

events.

4.2 Maximum horizontal displacements

The maximum displacements for single and for sequential ground motions, are shown in Fig.8 for various

characteristic cases. Permanent displacements are accumulated during oncoming earthquakes and therefore the


maximum displacements appear to be increased for the case of seismic sequences. These findings have been

observed for all the examined structures, for any case of siting (for irregular structures) and almost for the entire

set of the examined real seismic sequences under consideration.

Figure 7. Local damage index according to the Park-Ang model.

Finally, it can be easily concluded examining Fig. 8 that every structure under consideration may be excited

differently for each individual earthquake, or equivalently, a single earthquake from a specific seismic sequence

can be the most intense for a structure and less critical for another structure, but for the most of the cases, seismic

sequences generally cause more intense response in comparison with the ‘worst’ single ground motion.

4.3 Maximum residual displacements

The time history of horizontal displacements occurring at the top of the structures is shown in Fig. 9.

Residual displacements are accumulated during the seismic sequence. The influence of seismic sequences on the

residual displacements is clearly shown in Fig. 10, which presents selected results.

4.4 Interstorey drift ratio (IDR)

The interstorey drift ratio (IDR) is the maximum relative displacement between two successive stories

normalized to the storey height and it has been related to the structural or non-structural damage [12]. Figure 11

depicts characteristic examples for the IDR where it is evident that seismic sequences cause higher drifts in

comparison with the corresponding single ground motions.


Figure 8. Maximum horizontal displacements under single seismic events and sequential ground motions.

Figure 9.Time history of horizontal top displacement for 3-storey bldg under Mammoth Lakes earthquakes.

4.5 Residual interstorey drift ratio

Residual interstorey drift ratio (Res.IDR) has to do with the permanent deformation of a structure that

remains after a strong ground motion. Res.IDR evaluates the potential damage that the structure has sustained

[13, 14] after a strong ground motion. Figure 12 clearly shows that seismic sequences lead to increased residual

IDR.

4.6 Ductility demands

To quantify the severity of inelastic response, curvature ductility demand, φμ is used, which can be defined as

the ratio of ultimate curvature, uφ , to yield curvature, yφ , as

puφ

y y

φφμ 1

φ φ (3)

where pφ is the plastic curvature.

The inelastic response of 5-storey irregular building under the Mammoth Lakes seismic sequence is examined in

the following. Fig. 13 depicts the ductility demands both for single earthquakes and for seismic sequence

examining member No. 1 (ground floor column, see Fig. 4) and for the whole set of incident angles, i.e., for 0o,

90o, 180

o and 270

o, as mentioned in Section 2.4. It is evident that each incident angle has different effect in

comparison with the other cases under consideration. Furthermore, it is obvious that, in the most cases, the

multiplicity of earthquakes leads to increased ductility demands in comparison with the corresponding individual

seismic events.

1

2

3

0.00

0.02

0.04

0.06

0.08

0.10

0.12

MA1

MA2

MA3

MA4

MA5

MAT

Mammoth Lakes earthquakes / 3-storey reg. bldg (000)

Max. X

-dis

p. (m

)

Storey 1

2

3

4

5

0.00

0.02

0.04

0.06

0.08

0.10

0.12

MA1

MA2

MA3

MA4

MA5

MAT


Max. Y

-dis

p. (m

)S

tore

y

1

2

3

4

5

0.00

0.02

0.04

0.06

0.08

0.10

0.12

MA1

MA2

MA3

MA4

MA5

MAT

Mammoth Lakes earthquakes / 5-storey irreg. bldg (090)

Max. Y

-dis

p. (m

)

Sto

rey

1

2

3

45

0.00

0.01

0.02

0.03

0.04

0.05

IM1

IM2

IMT

Whittier Narrows earthquakes / 5-storey irreg. bldg (180)M

ax. Y

-dis

p.

(m)

Storey1

2

3

0.00

0.01

0.02

0.03

0.04

0.05

CH1

CH2

CHT

Chalfant Valley earthquakes / 3-storey irreg. bldg (180)

Max. X

-dis

p. (m

)

Storey1

2

3

0.00

0.01

0.02

0.03

0.04

0.05

CH1

CH2

CHT

Chalfant Valley earthquakes / 3-storey irreg. bldg (000)

Max.

X-d

isp. (m

)

Storey


Figure 10. Maximum residual displacements under single and sequential ground motions.

Figure 11. Maximum IDR under single and sequential ground motions.

Figure 12. Residual IDR for 5-storey irregular bldg under Mammoth Lakes earthquakes.

1

2

3

4

5

0.00

0.01

0.02

0.03

0.04

0.05

0.06

MA1

MA2

MA3

MA4

MA5

MAT


Max. Y

-Res. dis

p. (m

)

Sto

rey

1

2

3

45

0.00

0.01

0.02

0.03

0.04

0.05

0.06

MA1

MA2

MA3

MA4

MA5

MAT


Max. Y

-Res. dis

p. (m

)

Storey

1

2

3

45

0.00

0.01

0.02

0.03

0.04

CO1

CO2

COT

Coalinga earthquakes / 5-storey reg. bldg (000)

Max X

-Res.

dis

p. (m

)

Storey1

2

3

45

0.00

0.01

0.02

0.03

0.04

CO1

CO2

COT

Coalinga earthquakes / 5-storey irreg. bldg (000)

Max. Y

-Res.

dis

p.

(m)

Storey

1

2

3

45

0.000

0.002

0.004

0.006

0.008

0.010

MA1

MA2

MA3

MA4

MA5

MAT


Max. X

-Res.

IDR

(m

)

Storey1

2

3

45

0.000

0.002

0.004

0.006

0.008

0.010

MA1

MA2

MA3

MA4

MA5

MAT


Max. Y

-Res. ID

R (

m)

Storey


MA1

MA2

MA3

MA4

MA5

MAT

Figure 13. Ductility demands under single and sequential ground motions, for angles 0

o, 90

o, 180

o and 270

o.

5 CONCLUSIONS

This study examined the effect of sequential ground motions on regular and irregular three-dimensional RC

structures. Additionally, the siting configuration of the irregular buildings was investigated. Examining the

results of this study, important conclusions are found:

0

20

40

60

80

100

120

140

0

45

90

135

180

225

270

315

0

20

40

60

80

100

120

140

5-storey irreg. bldg

Member No. 1

MA1_earthquake

Ductility μZ(base)

Ductility μZ(top)

Ductility μY(base)

Ductility μY(top)

Du

ctilit

y d

em

an

ds

0

20

40

60

80

100

120

140

0

45

90

135

180

225

270

315

0

20

40

60

80

100

120

140


Member No. 1

MA2_earthquake

Ductility μZ(base)

Ductility μZ(top)

Ductility μY(base)

Ductility μY(top)

Du

ctilit

y d

em

an

ds

0

20

40

60

80

100

120

140

0

45

90

135

180

225

270

315

0

20

40

60

80

100

120

140


Member No. 1

MA3_earthquake

Ductility μZ(base)

Ductility μZ(top)

Ductility μY(base)

Ductility μY(top)

Du

ctilit

y d

em

an

ds

0

20

40

60

80

100

120

140

0

45

90

135

180

225

270

315

0

20

40

60

80

100

120

140


Member No. 1

MA4_earthquake

Ductility μZ(base)

Ductility μZ(top)

Ductility μY(base)

Ductility μY(top)

Du

ctilit

y d

em

an

ds

0

20

40

60

80

100

120

140

0

45

90

135

180

225

270

315

0

20

40

60

80

100

120

140


Member No. 1

MA5_earthquake

Ductility μZ(base)

Ductility μZ(top)

Ductility μY(base)

Ductility μY(top)

Du

ctilit

y d

em

an

ds

0

20

40

60

80

100

120

140

0

45

90

135

180

225

270

315

0

20

40

60

80

100

120

140


Member No. 1

MAT_earthquake

Ductility μZ(base)

Ductility μZ(top)

Ductility μY(base)

Ductility μY(top)

Du

ctilit

y d

em

an

ds


The displacements of the top level of the structures indicate strong increase under the effect of repeated

earthquakes and in some cases appear to be doubled or even more.

Under the influence of multiple earthquakes the residual displacements increase and accumulate.

The residual interstorey drift ratios are increased during sequential ground motions in comparison with

the more intense single ground motion.

Seismic sequences cause damage accumulation and in some cases, damage indices for structural

members can be increased fivefold or more in comparison with the ‘worst’ single earthquake.

Many structural members can behave elastically during single ground motions while during a seismic

sequence are damaged (inelastic behavior).

The ductility demands during multiple earthquakes are higher than those during the individual

earthquake. In some members, although that there is no need to endow them with adequate ductility

capacities during the single ground motion, in the case of seismic sequences the ductility demands

increase. For the seismic design of RC structures, the three-dimensional character of structural

modeling, of seismic loading as well as of response should be taken into account.

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