8
ELSEVIER PII: S0141-0296(96)00124-1 EngineeringStructures, Vol. 19, No. 7, pp. 568-575, 1997 © 1997 Elsevier Science Ltd All rights reserved. Printed in Great Britain 0141-0296/97 $17.00 + 0.00 Ductility spectra of reinforced concrete structures subjected to far-field seismic excitations Hong Hao and Thien-Cheong Ang School of Civil and Structural Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798 (Received November 1994; revised version accepted July 1996) This paper calculates the nonlinear ductility ratio spectra of low- to medium-rise RC structures subjected to simulated earthquake ground excitations. Two nonlinear structural models are used. One represents RC structural failure primarily by shear and the other represents the failure primarily by flexural tension and com- pression. Four types of earthquake ground motions are simulated for the calculation. They represent motions at a rock and a soft soil site caused by an earthquake of Richter magnitude 8 and epicentral distance 300 kin. For each type of motion, 20 time histories are simulated for the structural response analysis. The mean response spectra of ductility ratios and their coefficients of variation for the two structural models subjected to each type of motion are calcu- lated by using the 20 simulated time histories. The probability func- tions of the ductility ratios are formulated by using the mean values and coefficients of variation. These functions are used as prelimi- nary estimators of failure probabilities of RC structures. The results presented can be used to check the safety probability of low to medium rise RC structures under earthquake excitations. © 1997 Elsevier Science Ltd. All rights reserved. Keywords: ductility spectra, RC structures, earthquake excitations I. Introduction Since the 1985 Mexico earthquake, it has generally been realized by earthquake engineers and building authorities that structures located as far as 400 km away from an earth- quake epicentre may still be vulnerable to earthquake dam- age ~. There are many areas worldwide that are about 300- 400 km away from active faults but their current building codes do not include any seismic consideration in the designs. This raises many questions about the safety of structures. As more and more new buildings are constructed due to urbanization, it becomes an urgent problem to re- evaluate the safety of existing structures and to provide design guidelines for new structures which may be exposed to future, distant, large earthquakes. Current practice in seismic resistant design generally fol- lows the criteria that only minor structural damage is allowed for structures subjected to moderate earthquake conditions and that total structural damage or complete fail- ure should be prevented under severe earthquake con- ditions. These conditions can be fulfilled in deterministic design if earthquake ground motion records are available. However, any single record might not be able to represent the major statistical features of ground motions since there are many uncertainties involved even at a given site. More- over, in many areas which are several hundreds of kilo- metres away from active faults, there are no historical ground motion records available since earthquake threat has never been considered. Thus, using simulated ground motions in seismic resistant design becomes one of the common approaches and is stipulated by several seismic codes 2-5. There are also some advantages in using simu- lated motions in seismic response analysis since it is believed that the simulted motions can represent the major features of ground motions better than any single fully recorded time history. Besides, using simulated motions, the uncertainties that exist in ground motions can also be taken into account, thus allowing probabilistic studies to be conducted. It is recognized that uncertainties exist not only in ground motions but also in structures, thus the structural response prediction using a probabilistic approach is encouraged. 568

Ductility spectra of reinforced concrete structures subjected to far-field seismic excitations

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Page 1: Ductility spectra of reinforced concrete structures subjected to far-field seismic excitations

ELSEVIER PII: S0141-0296(96)00124-1

Engineering Structures, Vol. 19, No. 7, pp. 568-575, 1997 © 1997 Elsevier Science Ltd

All rights reserved. Printed in Great Britain 0141-0296/97 $17.00 + 0.00

Ductility spectra of reinforced

concrete structures subjected to

far-field seismic excitations Hong Hao and Thien-Cheong Ang

School of Civil and Structural Engineering, Nanyang Technological University, Nanyang Avenue, Singapore 639798

(Received November 1994; revised version accepted July 1996)

This paper calculates the nonlinear ductility ratio spectra of low- to medium-rise RC structures subjected to simulated earthquake ground excitations. Two nonlinear structural models are used. One represents RC structural failure primarily by shear and the other represents the failure primarily by flexural tension and com- pression. Four types of earthquake ground motions are simulated for the calculation. They represent motions at a rock and a soft soil site caused by an earthquake of Richter magnitude 8 and epicentral distance 300 kin. For each type of motion, 20 time histories are simulated for the structural response analysis. The mean response spectra of ductility ratios and their coefficients of variation for the two structural models subjected to each type of motion are calcu- lated by using the 20 simulated time histories. The probability func- tions of the ductility ratios are formulated by using the mean values and coefficients of variation. These functions are used as prelimi- nary estimators of failure probabilities of RC structures. The results presented can be used to check the safety probability of low to medium rise RC structures under earthquake excitations. © 1997 Elsevier Science Ltd. All rights reserved.

Keywords: ductility spectra, RC structures, earthquake excitations

I. Introduction

Since the 1985 Mexico earthquake, it has generally been realized by earthquake engineers and building authorities that structures located as far as 400 km away from an earth- quake epicentre may still be vulnerable to earthquake dam- age ~. There are many areas worldwide that are about 300 - 400 km away from active faults but their current building codes do not include any seismic consideration in the designs. This raises many questions about the safety of structures. As more and more new buildings are constructed due to urbanization, it becomes an urgent problem to re- evaluate the safety of existing structures and to provide design guidelines for new structures which may be exposed to future, distant, large earthquakes.

Current practice in seismic resistant design generally fol- lows the criteria that only minor structural damage is al lowed for structures subjected to moderate earthquake conditions and that total structural damage or complete fail- ure should be prevented under severe earthquake con- ditions. These conditions can be fulfilled in deterministic

design if earthquake ground motion records are available. However, any single record might not be able to represent the major statistical features of ground motions since there are many uncertainties involved even at a given site. More- over, in many areas which are several hundreds of kilo- metres away from active faults, there are no historical ground motion records available since earthquake threat has never been considered. Thus, using simulated ground motions in seismic resistant design becomes one of the common approaches and is stipulated by several seismic codes 2-5. There are also some advantages in using simu- lated motions in seismic response analysis since it is believed that the simulted motions can represent the major features of ground motions better than any single fully recorded time history. Besides, using simulated motions, the uncertainties that exist in ground motions can also be taken into account, thus allowing probabilistic studies to be conducted. It is recognized that uncertainties exist not only in ground motions but also in structures, thus the structural response prediction using a probabilistic approach is encouraged.

568

Page 2: Ductility spectra of reinforced concrete structures subjected to far-field seismic excitations

Ducti l i ty spectra o f RC structures: H. Hao and T.-C. Ang 5 6 9

Due to economic considerations, it is common practice that structures are not designed for zero risk of damage under severe earthquake conditions. Because of the cyclic nature of earthquake loading, the degree of damage sus- tained by a structure depends on many factors, one of which is the ductility ratio that characterizes the maximum struc- tural deformation. Other important factors include the load- ing history, the number and size of hysteretic loops formed in the response history, and the total energy dissipated by the member considered. Therefore, the ductility ratio only provides limited information about the behaviour of a struc- ture during an earthquake. However, the ductility ratio is simple in concept and easy to implement in design practice. As a result, it is widely used as a measure of the structural damage during earthquakes by both researchers and engin- eers 6'7. The ductility ratio is also employed in many seismic codes 2-5 as one of the control parameters in design. Hence, in this study it is employed as a preliminary estimator for the structural damage assessment.

The spectra of nonlinear response ductility ratios of low- to medium-rise RC structures are calculated. A site selected for the study is assumed to be 300 km away from a fault and no historical ground motion record is available. Ground motion time histories at the rock site are stochastically simulated for the analysis. The statistical attenuation model 8 for ground motions recorded on rock in Japan is chosen for the simulation. The local site effects are also considered by solving the site responses using the simulated motions at the rock site as the base rock input. Four types of motions are simulated to represent: the SH wave, the combined SH and Love wave on rock surface, and these two types of waves on the surface of a layered soil site as shown in Figure 1, where ' G W L ' stands for ground water level. Most seismic codes 2 5 require only three to four inde- pendently simulated motions for the structural response analysis, but more simulated motions can better represent the ground motion characteristics and produce a more reliable (less biased) probability prediction of structural responses. In this study, 20 time histories are simulated for each type of motion for use in the structural response analy- sis.

To model low- to medium-rise RC structural responses to earthquake excitations realistically, two nonlinear structural models are used. One represents the RC structural failure primarily by shear, the other represents the structural failure primarily by flexural tension and compresison. These two models are modified from the origin-oriented model and the trilinear stiffness degrading flexure model 6,9, respect- ively. Two ductility ratios that have been used to represent light and severe (but not collapsed) damage of RC struc- tures are 2 and 10 for the shear failure model, and 2 and 4 for the flexure compression and tension failure model 6.

The mean values and coefficients of variation of the duc- tility response ratio spectra of the two models subjected to the four types of motions are calculated by using the 20 simulated time histories. The probability and exceedance functions for the response ductility ratios are obtained by using the mean ductility ratios and their coefficients of vari- ation. The numerical results are presented and discussed. These results can be used to predict the probabilities of response ductility ratios for RC structures, which in turn can be used as preliminary estimators of the reliabilities of building structures subjected to earthquake excitations, and can also be used in the formulation of seismic resistant design guidelines.

depth (111)

0

1.15 silty clay Su=4.0KPa p=I4.0KN/m 3

sand Su=6.0KPa p=l 8.5KN/m 3 Dr=45%

sand Su=I5.0KPa p=l 9.6KN/rn 3 Dr=75qb 4.05

8.50 marine clay S~22.0KPa p=I6.3KN/m 3

17.40 silty clay Se=100.0KPa p=19.1KN/m 3

19.50 rnarineclay S~=128.0KPa p=lT:SKNhn 3

24.10 sihy clay Se=I33.0KPa p=l 9.SKN/m 3

28.80 silty clay S~=I45.0KPa p=20.SKN/m 3

31.20 sihy clay S,=I54.0KPa p=21.0KNBn 3

35.0 silty clay S~=169.0KPa p=21.0KN/m 3

36.20 silty clay Se=235.0KPa p=22.0KNhn3

38.0

39.80

41.40

44.20

46.10

49.80

52.0

53.0

60.0

Figure 1

)=22.0KN/m3 silty clay Sa=239.0KPa

silty clay S,=275.0KPa ~=22.0KN/m 3

clayey silty Se=322.0KPa ~=22.0KN/m 3

fine sand Su=366.0KPa )=22.0KN/m 3

3=23.0KNhn 3 coarse sand Su=385.0KPa

clayey silty S~-420.0KPa ~=23.0KN/m 3

clayey silty Se=450.0KPa ~=23.0KN/m 3

clayey silty S~500.0KPa )=23.0KN/m 3

base rock G--0.5GPa

Layered soil site

)=24.2KN/m 3

GWL

2. S i m u l a t e d g r o u n d m o t i o n s

It is common practice in earthquake engineering to use ground motion data from another area for an area in which no ground motion data is available, provided that both areas have similar geotechnical and tectonic conditions. For this reason, the statistical ground motion attenuation model of Japan 8 is chosen in this study. Based on the motions recorded on rock sites in Japan from 78 events, it was found that the horizontal peak ground acceleration attenuation fol- lows

logloPGA=O.44M- 1.381ogloX+ 1.04 cm/s 2 (1)

where M is the Richter scale and X is the epicentral distance in kilometres. The strong motion duration To is estimated by

logloTo = 0.31M - 1.20 s

and the velocity response spectrum is

IogloSv(M,X,T,E;) = 0 . 6 7 M - 1.191Og~oX- 1.85 +f~(M,T) -f2(X,T) - f3(~,T) cm/s

(2)

(3)

where T is the period, ~ is the damping ratio, fl(M,T) and f2(X,T) are the best fitting trial functions and f3(~,T) is a correction function for the damping effect.

Ground motions simulated in a form compatible with these statistical characteristics are used in this paper to rep- resent those on a rock site. The motions on the surface of the layered soil site are obtained by solving the site

Page 3: Ductility spectra of reinforced concrete structures subjected to far-field seismic excitations

5 7 0 Ductil i ty spectra o f RC structures: H. Hao and T.-C. Ang

responses using the simulated motions at the rock site as the base rock motion input. Since the above statistical data do not reveal any information on the ground motion fre- quency contents, it was demonstrated 1° that good motions can be simulated as stationary or quasistationary with dif- ferent frequency contents but compatible with the same ground motion attenuation laws. It was also proved ~° that by assuming different wave types in site response calcu- lations, the site amplification effects differ significantly.

In the present investigation, four types of ground motions have been simulated. They represent stationary SH wave motion on a rock surface (A), quasistationary combined SH and Love waves motion on a rock surface (B), and these two types of motions on the surface of the layered soil site (C and D, respectively). As stated earlier, an event of magnitude 8 at a distance of 300 km will be used to simu- late the motions on rock surface. It is calculated from equa- tions (1) and (2) that the strong motion duration is 19.8 s, peak acceleration is 13.4 cm/s z. Thus the motions are simu- lated with a duration of 20.48 s as the simulations are car- fled out in the frequency domain with a time increment 6t = 0.01 s. In this study, the central frequency and damping ratio of the Tajimi-Kanai jl ground acceleration power spectral density function for type A motion are 77r rad/s and 0.6, respectively. For type B motion, the SH wave win- dow is 0 - 1 0 s and the central frequency and damping ratio are also 7 ~- rad/s and 0.6, while the Love wave window is 9-20.48 s with the central frequency and damping ratio 27r rad/s and 0.2, respectively. The envelope function applied to the simulated motions are the Bogdanoff type ~2 with constants a = 0.206 and b = 0.0078. The simulated motions are also iterated to be compatible with the statisti- cal attenuating velocity response spectrum calculated from equation (3).

The motions on the surface of the layered soil site shown in Figure 1 are obtained by calculating the site responses. The incidence angle of the SH wave is assumed to be 90 °. The responses are calculated in the frequency domain. The soil nonlinearity is considered by using the equivalent strain method t3. The detailed information on obtaining the four types of motions has been given elsewhere ~°.

20 accelerograms for each type of motion were simulated for the analysis. Table 1 gives the mean values and standard deviations of the peak ground accelerations of the four types of motions. They are calculated as the ensemble averge of the 20 accelerograms. Figure 2 shows the typical accelerograms of the four types of motions. The mean velo- city response spectra with 5% damping and their coef- ficients of variation of the simulated motions are shown in Figure 3. The velocity response spectrum from equation (3) is also shown in Figure 3. From Table 1 and Figure 3 it can be concluded that the simulated motions do reflect the major statistical characteristics of the type of ground motion they represent since the mean peak accelerations

and velocity response spectrum correlate with the statisti- cal data.

It should be noted that the mean response spectra for the type A and B motions on rock are very similar even though their waveforms differ. This is because they are simulated to be compatible with the same attenuating response spec- trum. The response spectra for the two types of motions on soil surface differ considerably because the soft soil ampli- fication effects on different types of waves differ. As can be seen, the soft soil site amplifies the surface wave more than the SH wave. A more detailed discussion on site responses and amplification has been given elsewhere ~°.

3. Nonlinear structural models

Two nonlinear structural response models are used. One is the origin-oriented hysteretic model 6"9 shown in Figure 4, which is used to represent the structures with nonlinear deformation and failure characteristics controlled primarily by shear. The parameters P~,, P,.y, V~c and Vsy represent the reinforced concrete shear cracking strength, ultimate shear strength, the relative displacement produced by P,,,, and the relative displacement produced by P,:,., respectively. The hysteretic behaviour of this model occurs when the relative displacement is greater than V~( or less than -V,,. The reduction of load follows linear paths always directed through the origin, ie., AOA, A'OA' , or A'OA' . Oscillation can also follow linear paths such as A'OA" and A"OA" without developing hysteretic loops provided that the maximum displacement does not exceed the maximum dis- placement previously achieved.

The second model is the trilinear stiffness degrading hys- teretic model which is modified from the model used by other researchers 6'9. It is shown in Figure 5a, which is used to represent RC structures with nonlinear deformation and failure characteristics controlled primarily by flexure- induced tension and compression; where P ..... P,>., P,, and P,y represent the RC cracking and ultimate strengths by compression and tension, respectively. The corresponding relative displacement is denoted by V,.c, V,y, V,c and V,y. The hysteretic behaviour of this model occurs with every cycle of deformation when the load level is greater than P , or smaller than P,.. If the relative displacement is less than V~ or greater than V,.y, the model behaves exactly like a standard bilinear hysteretic model with stiffness k~ and k2. However, if the displacement is greater than V,y or less than VLy, a new bilinear hysteretic relation is formed. For example, if the maximum displacement is Vm~ correspond- ing to C in Figure 5a, the reduction of load follows the path CD with a slope rkt, where

V~ - V~, r - (4)

v,~,,~-v,,.

Table I Mean values and standard deviation of peak acceler- ations

Statistic Type A Type B Type C Type D

Mean (cm/s 2) 13.4 15.37 14.13 24.27 128.33 Standard 0.36 0.34 0.62 10.30 deviation (cm/s z)

As the load drops by P,.y - P,~ to reach point D, any further drop will follow the path DQ with a slope rk2.

The postyield bilinear hysteretic model is shown in Figure 5b. The origin of the skeleton curve is shifted to 0', which is the intersection point of line QC and the horizontal axis and equal to BC/2. The stiffnesses of the new bilinear model are rk~ and rk2, respectively. If the displacement is less than V,,,~ or greater than Vo,, a third bilinear model is formed similar to the second one.

The parameters in the two models can be obtained

Page 4: Ductility spectra of reinforced concrete structures subjected to far-field seismic excitations

Ductility spectra of RC structures: H. Hao and T.-C. Ang

8 0 -

4 0 -

~. O - E

~ - 4 - 0 -

8 0 -

4 0 -

< o E v ~ "-~ - 4 o

- 8 0 I I I I - 8 0 1 I I I 0 5 10 15 20 0 5 10 15 20

t ime(sec ) t ime(sec )

5 7 1

°lc 40

o - 4 0

- 80

°°to 40

o - 4 0

-BO I I I I I I 0 5 10 1,5 20 0 5 10 15 20

t ime(sec) t ime(sec)

Figure 2 Typica l s imu la ted g round acce le rograms for four types of mot ions : (a) s ta t ionary SH w a v e on rock; (b) quas is ta t ionary SH and Love waves on rock; (c) SH w a v e on soil surface; (d) SH and Love waves on surface

Figure 3

v

102. 1

10

o

b . . . . . .

/ e - - ~ Ip

10 -" -

• o l

10 -~-

/ , I i ,

1 0 - I ,, i i i r l l l l I i i i i i i l l I i i I l l l l l J I I I l l l l q 1 0 - 3 I I I I I I I p I I I I l l l l l I I I I I I I l l I I I I I l l l l I

10 -2 10 " 1 10 10 I 10 -2 10 "' 1 10 10 ~ freq(Hz) freq(Hz)

Mean ve loc i ty response spectra and the i r coef f ic ients of var ia t ion for four types of mo t ion

according to the stress-strain relations and the specifi- cations for RC structures. In this study, in order to be con- sistent with the building design, the parameters are obtained based on the RC stress-strain relation from BS81 l014. As the stress-strain relation is a property of the RC material, it does not vary much in different design codes. BS81 I 0 ta

was used because it is a nonseismic code, hence it is con- sistent with the assumption that seismic forces have never been considered in the building design practice in the area under consideration. Assuming a minimum member size of 200 x 200 mm and a maximum 400 × 1000 mm, and a concrete grade of 30 N/mm 2, which is commonly used in RC structures, the minimum, average and the maximum RC cracking strengths are obtained by assuming the code

specified nominal and the maximum allowable reinforce- ment, respectively, as in Table 2. The ultimate strengths are 1.6 times that of the corresponding cracking strengths and the ultimate displacement is 4 times that of the respective cracking displacement.

4. Dynamic response calculation

The ductility response spectra of the two nonlinear struc- tural models were calculated using the four types of ground motions as input and a 5% damping. The dynamic equation was solved in the time domain using step-by-step inte- gration. The Newmark method with the constant acceler- ation assumption was used for the integration as this

Page 5: Ductility spectra of reinforced concrete structures subjected to far-field seismic excitations

572 Ductility spectra of RC structures: H. Hao and T.-C. Ang

Psy

Psc

-Vsy -V

A"

A ~

A' -

O Vsc Vsy

-Psc

-Psy

Figure 4 Nonlinear shear failure model for RC structures

)

Table 2 RC structure cracking strengths

Minimum Average Maximum (kN) (kN) (kN)

Psc 70.0 300.0 700.0 Pcc 30.0 2880.0 6000.0 Pct 5.0 480.0 1000.0

method was found 7 to be the most stable for solving the nonlinear dynamic responses. During step-by-step inte- gration, the stiffness K changes according to the two struc- tural models. Whenever the stiffness change occurs, the integration time step is reduced to maintain the required accuracy.

The spectra of response ductility ratios are defined as I x = v ( t ) . . . . Ivy, for the shear failure model and as tx = v(t),~ax/v,, for the flexure tension and compression fail- ure model. The spectra are obtained with respect to the initial natural period of the single-degree-of-freedom struc- ture.

a: original

Pcy

Pcc

Q

B C

"Y #*

/ / / ~ ,,' rKI | a a

rK2 . - 'D

O Vcc Vcy Vmax

Ptc Pty

b: after yielding

Pc3,

Pcc

J f,2 )' Vcc' Vcy'=Vmax

Ptc

Pty"

Figure 5 Nonlinear flexure failure model for RC structures; (a) original; (b) after yielding

5. Probabi l i ty funct ion of ducti l i ty ratios

To establish the probabili ty function for the maximum response ductility ratios, the distribution of extreme struc- tural responses should be known. Previous studies revealed that the probabili ty distribution of extreme structural responses to a single earthquake follows the Gumbel type I distribution 15't6. This can be used to model the probabili ty distribution function for response ductility ratios 6,9 as

P(IX) = exp{-exp[-cKix - u]} (5)

where IX is the maximum response ductility ratio, c~ and u are parameters which depend on the mean value and stan- dard deviation of /x, respectively. If only 20 samples are available, c~ and u can be obtained by ~7

c~ = 1.063/~r, u = g - 0.493cr, (6)

where ~ and ~r, are the ensemble mean and standard devi- ation, respectively, of the 20 sample values of IX.

Using this function, the probabili ty of ductility ratios exceeding a certain level Ixo is

Q(ixo) = Probabili ty (IX > Ixo) = 1 - P(IXo) (7)

6. Numer ica l results

Figure 6 shows the mean spectra and coefficients of vari- ation of the shear failure model due to type A and B motions. It is noted that mean ductility ratios from the two types of motions are very close. However, the coefficients of variation vary. For the cases of P.~c > 300 kN, the duc- tility ratios are almost always smaller than one, implying the responses remain in the elastic range. This means such a distant large earthquake (M = 8 and distance = 300 km) will not cause any damage to structures located at rock sites. However, the ductility ratios are very large at small periods for structures with a member cross-section dimen- sion 200 × 200 mm and designed without reinforcement or only nominal reinforcement (P~, = 70 kN). But, as the per- iod increases, the response ductility ratio decreases rapidly.

Page 6: Ductility spectra of reinforced concrete structures subjected to far-field seismic excitations

Ductility spectra of RC structures: H. Hao and T.-C. Ang

10 . . . ~ ~ 1 -

o ~ ._o

- O

>' % ~ ""* "" \[~'~"~ L > .-, ~5 I 0 - ' (3

c 10 -~ % c •

o ~. ', '5

E "- 0

\ o

10 -2 ,,,,,,,,I ,',..,.,I '',-.,I ,,,,,,"I 10 -2 0 -2 10 -1 1 10 10 2 10 -2

Period (sec)

Figure 6 M e a n duc t i l i t y rat ios and coe f f i c ien ts o f va r i a t i on o f shear fa i lu re mode l . sub jec ted to t y p e A ( w i t h o u t star) and t y p e B (s tar red) m o t i o n s

a

b . . . . . . . . . . .

C

?,

z/ " "~/~/,X~t

I I I I I I I l I I r I I I I I I I I I I I I I I I I I I I I I I I I I

I0 --I 1 I0 I0 2 Period (sec)

Psi= 70 kN; . . . . . Psc = 300 k N ; - - -

0 -# >,

._-_ *5 -O

E

10

1

1 0 -I

1 0 -2_

-3 l 0 , , ,,..I I 0 -2 I 0 - i

x & \

i

c O

0 "C O >

0

0 9

C- O.) '5

0 0

i _

1 0 - I

I I l l l l i l I I I J t . . I I l i l l l t l I 1 0 - 2

I 10 10 2 10 - 2

Period (sec)

0

b C

%\

/ \ ~ I \ V / \1

I I I I I f l l I I I I I I I I I I I I I I I I I I 1 I I I I I I l l I

10 -~ 1 10 10 2 Period (sec)

Figure 7 M e a n duc t i l i t y rat ios and coef f i c ien ts o f va r i a t i on o f shear fa i lu re m o d e l . - sub jec ted to t y p e C ( w i t h o u t star) and t y p e D (s tar red) m o t i o n s

P=c = 70 kN; . . . . . Psc = 300 kN; - - -

5 7 3

P~ = 700 kN

P~ = 700 kN

Thus structures built on rock sites are generally safe from such distant large earthquakes if they are properly designed. However, structures with short vibration periods, and with members of cross-section dimension 200 x 200 mm are vul- nerable if not adequately reinforced.

It is worth noting that the coefficients of variation are rather small when the response is elastic, but they increase as inelastic response occurs ( / x > 1). The differences between the coefficients of variation due to type A and type B motions are also relatively larger when inelastic response occurs. This is because of the frequency difference between the two types of motions. Since the velocity response spec- tra of the two types of motions are consistent, the elastic responses of structures subjected to those motions will also be consistent. However, once yielding occurs, the inelastic responses will differ as the two types of motions have dif- ferent frequency contents. Actually, if the frequency vari- ation of type B quasistationary ground motion coincides with the change of vibration frequencies due to the change

in structural stiffness, severe inelastic responses will be expected.

Figure 7 shows the same results obtained by using type C and type D motions. It can be seen that ductility ratios due to type C motion, which represents motion on the sur- face of the soil site due to vertically propagating SH wave, are almost always less than 1 for structures with Psc = 700 kN, less than 6 for Psc = 300 kN, but are very large for P~. = 70 kN. This means that if structures are designed with adequate member dimensions and reinforce- ment, no severe damage (/z = 10) will occur. However, the ductil i ty ratios due to type D motion are very large at small periods which correspond to severe damage. The type D motion represents ground motion on the surface of the soil site due to the combined SH and Love waves. Thus, if the surface wave is present, damage is expected for structures on soft soil even if the structures are designed with the maximum allowable reinforcement. Nevertheless, the duc- tility ratios for Psc > 300 kN are less than 10 if the period

Page 7: Ductility spectra of reinforced concrete structures subjected to far-field seismic excitations

5 7 4 Ductility spectra of RC structures: H. Hao and T.-C. Ang

o

2

m

o

-(D

c- o ©

E

1 0 -

I

10 -1

1 0 -2

\ E o

o 'E

~- 10 - ' J

\ - c

",,,~.~_.

~ 0 0

b . . . . . . . . . . .

C

\ I t i l l l l t I I i l r l l l l I I I l l l l l q 1 0 - 2 I l i l i l t l I I l i l [ I } l I I I I I I I I I I i I I I I I I I 1

10-1 1 10 102 10-2 10-1 1 10 102 Period (sec) Period (sec)

Figure 8 M e a n duc t i l i t y rat ios and coef f i c ien ts of va r i a t i on o f f l exu re fa i lu re mode l . - - P¢c = 30 kN, Pct= 5 kN; . . . . . Pcc = 2880 kN, Pc,= 480 kN; - - - Pc~= 6000 kN, P~t = 1000 kN sub jec ted to t y p e A ( w i t h o u t star) and t y p e B (s tar red) m o t i o n s

T > 1 s. This means that no severe damage will occur to properly designed structures if their dominant vibration per- iods are larger than 1 s.

The mean ductility ratios and their coefficients of vari- ation for the flexural failure model to types A and B and types C and D ground motions are shown in Figures 8 and 9, respectively. It also can be seen that if structures are built on a rock site and designed with adequate reinforcement, no yielding will occur when they are subjected to type A and B motions. Yielding might occur but structures will not suffer even minor damage (/~ = 2) when they are subjected to type C motion. In the case of type D motion, severe damage is expected for stiff structures even if they are designed with adequate reinforcement and member dimen- sions, but no severe damage (/_~ -> 4) will be expected for flexible structures with periods T > 1 s and designed with proper reinforcement.

The probabili ty of occurrence of certain ductility ratios

to each type of motion can be calculated using the results presented in Figures 6-9. For example, for a structure hav- ing natural period T = 1 s and subjected to type C motion, if its member ' s shear cracking strength is Psc = 300 kN, and the compression and tensile strengths are P(c = 2880 kN and Pc, = 480 kN, respectively, from Figures 7 and 9, the mean ductility ratios and coefficients of variation can be obtained. Using equation (6), the parameters a and u are calcualted to be 15.75 and 1.47 for the shear failure model, and 75.79 and 0.89 for the flexural failure model, respect- ively. Thus the probabili ty and exceedance functions can be established. The plots of P(/.t) and Q(/~) for these cases are shown in Figure 10.

For structures with other values of P~c in the shear failure model, or Pcc and P,c in the flexure failure model, the mean values and coefficients of variation of the ductility ratios can be similarly calculated or interpolated from the results presented in Figures6-9. Thus the probabili ty and

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10 -2 1 0 -1 1 10 102 10 -2 1 0 -1 1 10 1 0 2 Period (sec) Period (sec)

Figure 9 M e a n duc t i l i t y rat ios and coef f i c ien ts o f va r ia t i on o f t he f l exu re fa i lu re mode l . - - Pcc = 30 kN, Pc, = 5 kN; . . . . . Pcc = 2800 kN, Pct= 480 kN; - - - Pcc= 6000 kN, Pct= 1000 kN sub jec ted to t y p e C ( w i t h o u t star) and t y p e D (s tar red) m o t i o n s

Page 8: Ductility spectra of reinforced concrete structures subjected to far-field seismic excitations

Ductility spectra of RC structures: H. Hao and T.-C. Ang 575

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Probab i l i t y and exceedance f unc t i ons o f duc t i l i t y rat ios fo r s t ruc tu re w i t h per iod T = 1 s s u b j e c t e d to t y p e c m o t i o n . (a) , P ( u ) ; - - - O(u)

exceedance functions of the ductility ratios for structures having different strengths and periods can be similarly established.

7. Conclusions

A method for calculating the spectra of response ductility ratios for two structural models has been presented, The two models represent low- to medium-rise reinforced con- crete structures failing primarily by shear and by flexure compression and tension, respectively. The method has been used to analyse the responses of RC structures in an area 300 km away from the epicentre of a magnitude 8 earthquake of the Richter scale. The mean values of the ductility ratios and coefficients of variation of the two struc- tural models subjected to four types of simulated ground motions were calculated and discussed. It was found that the structures, whether built on rock or on the surface of a layered soil site, are generally safe if they are designed with adequate member dimensions and reinforcement. The results were also used to establish the probability and exceedance functions for response ductility ratios. These can be used as a preliminary probabilistic measure of build- ing safety and as a probabilistic seismic resistant design reference. The numerical results presented can also be used for RC structures in other areas, provided that they have the same basic design criteria and earthquakes with the same statistical nature as those used in this investigation.

References

1 Cassaro, M. A. and Romero E. M. (Eds) 'The Mexico earthquake, 1985: factors involved and lessons learned', ASCE, New York, 1987

2 Associate Committee on the National Building Code, 'NBCC, National building code of Canada, 1985', National Research Council of Canada, Ottawa, Ontario, 1985

3 Gomez, R and Garcia-Ranz, F. (translators), 'The Mexico earthquake of September 19; 1 9 8 5 ~ o m p l e m e n t a r y technical norms for earth- quake resistant design, 1987 edition', Earthquake Spectra, 1988, 4 (3), 441-459

4 Standards Association of New Zealand 'Code of practice for general structural design and design loadings for buildings: NZS 4203:1992', Wellington, New Zealand, 1992

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6 Murakami, M. and Penzien, J. 'Nonlinear response spectra for proba- bilistic seismic design and damage assessment of reinforced concrete structures', Report UCB/EERC-75/38, University of California at Berkeley, CA 1975

7 Allahabadi, R.- 'Drain-2DX, Seismic response and damage assess- ment for 2D structures', PhD dissertation, University of California at Berkeley, CA, 1987

8 Watabe, M. 'Characteristics of earthquake ground motions', paper presented in seminar, Department of Civil Engineering, University of California at Berkeley, CA, 1988

9 Clough, R. W. and Penzien, J. Dynamics of structures (2nd edn) McGraw-Hill, New York, 1993

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11 Tajimi, H. 'A statistic method in determining the maximum response of a building structure during an earthquake', Proc. 2WCEE, Vol. 2 Tokyo, 1960, pp. 781-798

12 Bogdanoff, J. L. 'Response of a simple structure to random earth- quake type disturbance', Bull. Seis. Soc. Am. 1961, 51, 293-310

13 Schnabel, P. B., Lysmer, J. and Seed, H. B. 'SHAKE, A computer program for earthquake response analysis of horizontally layered sites', Report UCB/EERC-72-12, University of California at Berke- ley, CA, 1972

14 British Standard, 'Structural use of concrete,' BS8110, British Stan- dards Institution, London, 1985

15 Ruiz, P. and Penzien, J. 'Artificial generation of earthquake accelero- grams', Report UCB/EERC-69-03, University of California at Berke- ley, CA, 1969

16 Penzien, J. and Liu, S. C., 'Nondeterministic analysis of nonlinear structures subjected to earthquake excitations', Proc. 4th World Con- ference on Earthquake Engineering, Santiago, Chile, 1969, pp. 114-129

17 Gumbel, E. J. Statistics of extremes, Columbia University Press, New York, 1958