Damage index for seismic analysis of RC members

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A Damage Index for the Seismic Analysis of Reinforced Concrete MembersMario E. Rodrigueza; Daniel Padillaa

a National University of Mexico, Instituto de Ingenieria, Mexico City, Mexico

To cite this Article Rodriguez, Mario E. and Padilla, Daniel(2009) 'A Damage Index for the Seismic Analysis of ReinforcedConcrete Members', Journal of Earthquake Engineering, 13: 3, 364 — 383To link to this Article: DOI: 10.1080/13632460802597893URL: http://dx.doi.org/10.1080/13632460802597893

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Journal of Earthquake Engineering, 13:364–383, 2009

Copyright � A.S. Elnashai & N.N. Ambraseys

ISSN: 1363-2469 print / 1559-808X online

DOI: 10.1080/13632460802597893

A Damage Index for the Seismic Analysisof Reinforced Concrete Members

MARIO E. RODRIGUEZ and DANIEL PADILLA

Instituto de Ingenieria, National University of Mexico, Mexico City, Mexico

This article proposes a damage index for the seismic analysis of Reinforced Concrete membersusing the hysteretic energy dissipated by a structural member and a drift ratio related to failure inthe structure. The index was calibrated against observed damage in laboratory tests of 76 RCcolumn units under various protocols. Values obtained in this calibration had acceptable agreementwith the levels of damage observed in the test specimens. An analysis of the parameters involved inthe definition of the proposed damage index shows the importance of displacement history in thedrift ratio capacity of structures.

Keywords Seismic Damage; Damage Index; Seismic Analysis; Reinforced Concrete Elements

1. Introduction

A measure of structural damage for a postulated earthquake is relevant for the seismic

analysis of both new and existing building and bridge structures. Several damage indices

have been proposed in the literature to quantify these measures. With respect to the

existing structures, such damage indices provide important information that could be

implemented in the initial assessment and retrofit decision-making process. They can

also be used for performance-based engineering approaches. Such indices have been

reviewed in the literature [Cosenza et al., 1993; Williams and Sexsmith, 1995; Ghobarah

et al., 1999; Teran-Gilmore and Jirsa, 2005], evidently stressing the need for better

damage indices. Several damages indices have been recently proposed. Erduran and

Yakut [2004] proposed a damage measure expressed in terms of interstory drift ratio and

the effect of displacement history is not taken into account. Colombo and Negro [2005]

proposed a damage index defined as the ratio of the initial and the reduced resistance of a

structure, and requires the definition of several parameters related to ductility and energy

dissipation. Kim et al. [2005] proposed a damage measure based on results of finite

element analyses, in which material models were modified to consider fatigue damage

based on results of numerical tests. In the present article, only the damage index of Park

and Ang [1985] is discussed because quantities involved in that index are also involved

in the damage index later proposed.

Park and Ang [1985] proposed the damage index, IPA, that is widely used in the literature.

This index is perhaps one of the earliest and the most popular damage indeces defined as

IPA ¼um

uu

þ �R

d EH

m ry uu

(1)

Received 2 July 2007; accepted 2 July 2008.

Address correspondence to Mario E. Rodriguez, Instituto de Ingenieria, National University of Mexico,

Mexico City, Mexico; E-mail: [email protected]

364

Downloaded By: [UNAM] At: 23:11 4 August 2010

where parameter um is the maximum displacement of a SDOF system responding to an

earthquake and uu is the ultimate displacement under monotonic loading. Parameters EH

and ry are the hysteretic energy dissipated by the SDOF system and the yield resistance of

this system, respectively. Parameter b considers the effect of repeated loading. It is

assumed that this system has a mass m, and a maximum displacement ductility ratio

equal to mm. The natural circular frequency of this system, o, is equal to

! ¼ffiffiffiffik

m

r(2)

where k is the lateral stiffness of the elastic system.

Some disadvantages of using index IPA have been discussed in the literature [Cosenza

et al., 1993; Williams and Sexsmith, 1995; Ghobarah et al., 1999]. Among these disadvan-

tages, it is noted that the index IPA is not normalized as it is not equal to 0 when um is equal to

uy, where the latter parameter is the structural displacement at yielding, and the index IPA is

not equal to 1 when the structure fails under monotonic loading.

2. Proposed Seismic Damage Index

The proposed damage index has evolved from the seismic damage parameter introduced by

Rodriguez [1994]. This parameter considers a level of acceptable seismic performance. In

fact, such a parameter was calibrated using several earthquake records and a specific level of

seismic performance not severe enough to cause a collapse. Results obtained using that

parameter were found consistent with observed seismic damage in 11 earthquakes experi-

enced in the past in different countries [Rodriguez and Aristizabal, 1999].

It is known that the ratio drift to height is relevant in the seismic behavior of structures.

As shown later, the proposed damage index is calibrated against experimental results using

measured drift ratios in lateral load tests of RC column units. It follows that it is highly

useful to express a damage index in terms of drift ratio. This can be done by expressing the

displacement u, involved in the derivation of Eq. (1), in terms of drift ratio, y, that is

� ¼ u

h(3)

where h is the column height.

Although, in this article, the authors do not address how to extend their findings to the

case of regular multistory buildings, the following derivation of the proposed damage index

has also been used for assessing seismic damage in existing multistory buildings [Rodriguez

and Padilla, 2006]. It is useful to express the restoring force r(t) of the SDOF system as a

function of the restoring base overturning moment of the system, M(t), that is:

rðtÞ ¼ MðtÞh

: (4)

For the elastic response, the moment M(t) is given by

MðtÞ ¼ k� �: (5)

From Eqs. (4) and (5), and the definition of k we obtain:

k� ¼ k h2: (6)

From Eqs (2) and (6) we obtain an expression for o in terms of ky:

! ¼ffiffiffiffiffiffiffiffik�

mh2

r: (7)

A Damage Index for the Seismic Analysis of Reinforced Concrete Members 365


The proposed damage index, Id, for the SDOF system previously described is defined as

Id ¼EH

E�; (8)

where parameter El (shown in Fig. 1) is defined as

E� ¼ k� �2c : (9)

In the above expression, parameter yc is the maximum drift ratio in an elastic SDOF (with

the same frequency of the analyzed system) such that the energy absorbed in this elastic

system, El, is equal to the hysteretic energy at collapse (EH = El). In the seismic demand-

to-capacity problem, parameters EH and El would correspond to the demand and capacity

terms, respectively. Therefore, if the plastic work demand is smaller than the capacity,

then according to Eq. (8), Id < 1.

From Eqs. (8) and (9) we obtain

Id ¼EH

k� �2c

: (10)

Another interpretation of parameter yc is given in the following. At collapse (EH = El)

that is when Id = 1, from Eq. (10) we obtain

�2c ¼

E�

k�: (11)

Since ky is the elastic energy absorbed by the SDOF system when it reaches the rotation

y = ±1 (see Fig. 1), �2c can be interpreted as the fraction of this energy that is dissipated

by the nonlinear system at collapse (EH = El) in the form of plastic work. By analyzing

results from a column database, it is shown later that parameter �2c appears to be constant

for RC members with similar structural characteristics, regardless of the displacement

history applied up to collapse. It follows that since �2c is the dimensionless plastic work

capacity in an RC member, see Eq. (11), for RC members with similar structural

characteristics, this capacity can be approximately considered an invariant property.

A second form of the proposed seismic damage index, Id, is proposed using a dimen-

sionless form of parameter EH, which is expressed with parameter g, and is computed from

[Rodriguez, 1994]

� ¼

ffiffiffiffiffiEH

m

q! �m h

; (12)

θ

M Kθ

θm1 θc θm2 1

θc θm1θm21

Eλ

FIGURE 1 Parameters involved in the definition of the proposed damage index.

366 M. E. Rodriguez and D. Padilla


where parameter ym is defined as:

�m ¼um

h: (13)

It is a matter of interest that parameter g has also been used by Fajfar [1992] along with a

seismic design procedure which addresses the effect of cumulative seismic damage.

Parameter g has been evaluated for a set of ground motions and was found that it is

dependant on structural and ground motion parameters [Fajfar, 1992].

From Eqs. (7) and (12) we obtain:

�2 �2m ¼

EH

k�: (14)

The above expression indicates that �2 �2m is equal to the dimensionless plastic work

demand in the SDOF system. Since k� �2m is the energy absorbed in the elastic system

(with the same frequency of the analyzed system) when it reaches a maximum drift ratio

equal to ym, Eq. (14) indicates that when a nonlinear system reaches collapse (EH = El) at

a drift ratio ym, it will dissipate an hysteretic energy at collapse that is equal to g 2 times

that elastic energy. Furthermore, at collapse (EH = El), from Eqs. (11) and (14) we obtain:

�c ¼ � �m: (15)

To illustrate Eq. (15), let us analyze two cases of a SDOF system that reaches collapse (EH =

El) at either ym = ym1 or ym = ym2. For ym = ym1, see Fig. 1, �m � �c, and therefore � � 1,

which indicates that in the system the hysteretic energy at collapse (EH = El, see shaded area

in Fig. 1) is larger than the elastic energy absorbed in the elastic system when it reaches the

maximum drift ratio ym1 (see Fig. 1). For ym = ym2, see Fig. 1, �m � �c, and therefore � � 1,

which indicates that the hysteretic energy at collapse (EH = El) is smaller than the energy

absorbed by the elastic system when it reaches the maximum drift ratio ym2 (see Fig. 1).

Finally, a relationship between Id and g can be obtained by combining Eqs. (10) and

(14), after which we obtain:

Id ¼� �m

�c

� �2

: (16)

For the case EH < El, using Eqs. (11) and (14) leads to � �m � �c, which in Eq. (16)

implies once again that Id � 1 .

It is of interest that according to the definition of Id, see Eq. (16), knowing the plastic

work capacity of a RC member (related to parameter yc) and the expected shape of the

hysteresis loops, we could predict whether or not a RC member would fail for a target

displacement history and deformation ym. This can be done by comparing the plastic

work capacity (k� �2c , see Eq. 11) and the plastic work demand (k� �

2 �2m, see Eq. 14) for

the target displacement history.

Usually, Force-Displacement relationships are used for the seismic analysis of SDOF

systems. However, the above derivation, based on Moment-Rotation relationships could

be also used in the Force-Displacement domain. These can be done considering that

parameter EH computed in either of the two mentioned domains are equal. This is due to

the fact that the incremental plastic work using the former type of relationship, DF Du, is



equal to the incremental plastic work using the latter relationship, DM Dq. For example, if

we want to express Eq. (11) in the Force-Displacement domain, using the given defini-

tions of parameters y and ky, this equation can be expressed as

u2c ¼

E�

k(17)

where uc is equal to yc h.

3. Description of the Column Database

The parameters involved in the definition of the proposed damage index were evaluated

using an experimental database consisting of a set of different RC columns tested in a

laboratory under axial and cyclic lateral loading. This column database is described in the

following. The experimental database used in this study for the calibration of Id consisted

of 76 RC columns tested in the last 30 years in different laboratories of USA, Japan, New

Zealand, and Canada. Figure 2 shows three different types of column test setups found in

the column database. A detailed description of the database can be found in Rodriguez

and Padilla [2006]. Only a brief description of this database is given in the following.

Table 1 lists several characteristics of the columns of the database such as compressive

strength of concrete f’c; tensile stress at yielding of longitudinal and transverse reinforce-

ment, fy and fyt, respectively; axial load ratio P/Ag f’c, where P is the axial load and Ag is

the column section area; ratio of longitudinal reinforcement, rl; ratio of volume of

transverse reinforcement to volume of column core, rt ; and ratio rt /rACI, where rACI

is the amount of transverse reinforcement prescribed by Ch. 21 of the ACI 318-05 [2005].

Table 1 also lists the column height, h (in columns with single or double curvature, see

Fig. 2), the shear span ratio M/VD, where M and V are the maximum flexural moment and

shear acting in a critical column section, respectively, and D is the depth of the column

cross section. In addition, Table 1 shows some test results such as the lateral stiffness of

the test unit, k; maximum drift ratio reached during testing at an ultimate level of damage

later described, ym; and hysteretic energy, EH, computed up to that drift ratio, with a

procedure which is also described later. The initial elastic lateral stiffness in column test

units, k, was obtained from an envelope of the measured lateral load-deformation hyster-

esis loops assuming a bilinear inelastic rule and a secant initial lateral stiffness at about

3/4 of the maximum measured lateral force.

Table 2 lists a summary of several characteristics of columns of the database such as

section type, axial load ratio, type of observed failure, type of lateral loading, and whether

h

P

P

h

P

h

h

(F, u)(F, u)

(F, u)(F, u

a) Single curvature b) Double curvature c) Single curvature with center stub

FIGURE 2 Types of test setups in the column database.



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372


the columns had low or high confinement. In addition, Table 2 lists the range of values

for typical design parameters in these columns, such as compressive strength of concrete

f’c, tensile stress at yielding of longitudinal and transverse reinforcement, fy and fyt,

respectively; axial load ratio P/Ag f’c; ratio of volume of transverse reinforcement to

volume of column core rt; confinement index (rt fyt/f’c); maximum drift ratio ym; and

shear span ratio M/VD. The range of values for this parameter and the observation of

damage at end of testing for the column units of the database suggest that their failure

mode were either flexure or flexure-shear dominated.

Figure 3 shows typical examples of lateral displacement histories applied in the 76

RC columns of the database. As seen here, these cyclic displacement histories cover a

wide range of displacement histories. Most column units of the database were tested

TABLE 2 Summary of characteristics of columns of the database

Characteristic Quantity Characteristic Range

Selected test specimens 76 f’c (Mpa) 19.8–115.8

Circular test specimens 24 fy (MPa) 240–511

Square test specimens 52 fyt (MPa) 255–792

Test specimens with detailed description

of damage

23 P/Ag f’c 0–0.9

Test specimens w/o detailed description

of damage

53 rt 0.001–0.039

Test specimens with rt fyt / f’c > 0.1 36 rt / rACI 0.17–.49

Test specimens with rt fyt / f’c < 0.1 40 rt fyt / f’c 0.02–0.5

Test specimens with P/Ag f’c < 0.1 29 ym (%) 0.5–10.8

Test specimens with 0.1< P/Ag f’c < 0.3 26 M/VD 1.0–10

Test specimens with P/Ag f’c > 0.3 21

Test specimens with monotonic loading 2

Test specimens with reversed cyclic lateral

loading

63

Test specimens with earthquake type loading 11

–400

4080

120

0–80

–40

0

0

0

4080

Δ (

mm

)

–90–45

04590

0

-120-60

060

120

Δ (

mm

)

–400

4080

120

0–40

04080

120

0

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–20

0

20

40

Δ (

mm

)

–20–10

01020

0

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mm

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(m

m)

Δ (

mm

)

–150–75

075150

0

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mm

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m)

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mm

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0

6

12

0

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-150-75

0

75

150

0

Δ (

mm

)

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0

10

20

0

Δ (

mm

)

FIGURE 3 Lateral displacement histories in the test specimens of the database.



statically and in other cases dynamically using earthquake excitations such as column

units No 21–26 (see Table 1).

4. Definition of Parameters Involved in Proposed Damage Index

4.1. Definition of Parameter qm

A significant factor pertaining to the evaluation of the database needs mentioning here. The

definition of failure considered in this study corresponded to the response of a structural

element at an ultimate drift ratio equal to ym, which was defined when the lateral strength of

the test unit had a decay of 20%. Beyond this level of strength decay, total repair is

generally needed [Park and Ang, 1985]. This definition of ym is illustrated in Fig. 4. It is a

matter of great interest that in most of the database selected in this study, testing was

terminated at levels of strength decay of about 20%. In fact, only 10 test specimens of the

database reached levels of strength decay of 50% or more, which would more likely

represent a collapse or an imminent collapse. With the given definition of ym, a set of

values for this parameter was generated using the experimental database listed on Table 1.

Quite interestingly, different authors have proposed different definitions for classifi-

cation of damage. For example, values found using the well known Park and Ang index

fell in a wide range of damage classification. According to this index a threshold value

between repairable and nonrepairable damage is 0.4, leading to a wide range of classi-

fication of severe damage before collapse from 0.4–1.0 [Park et al., 1985]. In fact, it is

also suggested that the value of 1.0 representing collapse in the Park and Ang index

should change to a value of 0.8 to represent collapse [Williams and Sexsmith, 1995].

4.2. Definition of Parameter EH

The hysteretic energy, EH, for the 76 test specimens of the database was computed by

numerical integration of the lateral force-drift hysteretic response cycles recorded in these

test specimens [Rodriguez and Padilla, 2006]. For each test specimen, parameter EH was

computed up to the cycle with the drift ratio equal to ym, with the additional consideration

that if a second cycle at that drift ratio dropped its strength to more than 20%, then that

cycle was not computed in the evaluation of EH.

4.3. Definition of Parameter qc

Parameter yc for the database was computed using Eq. (11), making El = EH and computing

EH as described in Sec. 4.2, and ky was computed using Eq. (6) with the definition of k given

in Sec. 3.

F

θm

0.8Fmax

Fmax

θ

FIGURE 4 Definition of maximum drift ratio ym.



5. Evaluation of Parameters qc, qm, and g Using the Experimental Database

From the database, seven groups of test specimens were used for a detailed evaluation of

parameter yc. As shown in Table 3, each of these groups had similar structural properties,

defined with the axial load ratio P/Ag f’c, ratio rt/rACI, and shear span ratio M/VD.

However, specimens in each group were subjected to different displacement histories.

In some cases, within each group, parameters k and h, which define ky (see Eq. 6), had

different values (see Table 1). For these seven groups of test specimens, parameter g was

evaluated at the maximum drift ratio ym reached using Eq. (12) and results are shown in

TABLE 3 Analysis of the effect of displacement history on drift ratio capacity of similar

RC column

Group Test Unit g ym yc

(a) LEH1015 1.08 0.099 0.107 P/Agf c < 0.1

LEH815 1.09 0.091 0.100 82% <t /rACI < 94%

KOWAU1 1.67 0.062 0.103 5.3 < M/VD < 10

KOWAU2 0.96 0.108 0.104

(b) KUN7 1.14 0.059 0.067 P/Agf c < 0.1

KUN8 1.31 0.057 0.075 100% < rt /rACI < 126%

KUN9 1.18 0.066 0.078 M/VD = 4.5

KUN10 1.21 0.066 0.079

KUN11 1.32 0.055 0.073

KUN12 1.80 0.039 0.07

(c) KANSTC1 1.15 0.0461 0.0532 P/Agf c < 0.1

OHNO84L3 1.24 0.0456 0.0564 46% < rt /rACI < 76%

SOES86U1 0.81 0.0612 0.0498 3.1 < M/VD < 5

TP001 1.5 0.034 0.051

TP002 1.07 0.044 0.047

TP005 0.71 0.0724 0.051

(d) ANG81U4 1.05 0.0365 0.0383 0.2 < P/Agf c < 0.23

TANA90U1 0.96 0.0401 0.0386 276% < rt /rACI < 369%

TANA90U4 0.72 0.0487 0.0349 M/VD = 4

TANA90U2 0.80 0.0404 0.0322

ZAHN86U7 0.79 0.0516 0.0407

(e) ZHO22309 0.58 0.0383 0.0224 P/Agf c > 0.33

ARA82102 0.85 0.0340 0.0290 199% < rt /rACI < 549%

WAT89U9 1.14 0.0218 0.0248 3 < M/VD < 4

ANG81U3 0.91 0.0318 0.0290

ZHO1248 0.87 0.0366 0.0318

(f) SOES86U2 0.69 0.0314 0.0218 P/Agf c = 0.3

SOES86U3 0.71 0.0282 0.0200 40%<rt /rACI < 111%

SOES86U4 0.94 0.022 0.0207 M/VD = 4

(g) WAT89U6 0.98 0.0157 0.0154 P/Agf c > 0.5

WAT89U7 1.27 0.0084 0.0106 34% < rt /rACI < 103%

WAT89U8 1.15 0.0111 0.0127 M/VD = 4



Table 3. Calculated values of parameter yc using Eq. (11) are also shown in Table 3 and

plotted in Fig. 5. As seen there, for each group of test specimens, parameter yc is nearly

constant, which leads to two important observations. The first observation is that speci-

mens with similar structural properties (as listed above) subjected to different loading

protocols have different drift capacities (measured as ym) and an approximately constant

equivalent elastic drift yc. The second observation is that according to the definition of

parameter yc, in columns with similar structural properties subjected to different loading

protocols, the plastic work capacity appears to be constant (see Fig. 5), and this capacity

can be evaluated as k� �2c , see Eq. (11).

The line at 45o with the x axis in Fig. 5 indicates the case yc = ym . Therefore, the zone

above the line corresponds to the case yc > ym, and the zone below the line corresponds to

the case yc < ym. Accordingly, the former and later cases correspond to the cases g > 1 and

g < 1, respectively, as seen in Eq. (15). These results also give an insight into the effect of

displacement history on the ultimate deformation capacity of an RC element, which is

accounted for by parameter g. For example, it can be said that the condition g = 1.0 would

be related to a medium effect of displacement history, and g > 1.0 and g < 1.0 would be

related to a high and low effect of displacement history, respectively.

Figure 6 shows a plot of calculated values of parameter �2c obtained using Eq. (11) for

the 76 test specimens of the database. These values are plotted on the y axis as a function

of ym, which is plotted on the x axis. Results shown in Fig. 6, and Eqs. (11) and (15),

indicate that the parabolic curve y = x2 defines the condition yc = ym or g = 1.0. With the

same reasoning, it can be shown that results above the parabolic curve would correspond

to the condition yc > ym and g > 1.0. Results below the parabolic curve would correspond

to the condition yc < ym and g < 1.0.

The above discussion gives ground to explain what has been commonly observed in

cyclic lateral load tests of similar specimens. A hysteretic energy at the ultimate damage

level dissipated with a large number of lateral load cycles would lead to less deformation

capacity than in the case when a hysteretic energy is dissipated in a similar specimen with

a smaller number of cycles at the same damage level. To elaborate on this finding, results

obtained in six identical RC column units, tested under various protocols [Takemura and

Kawashima, 1997] are discussed in the following.

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14

θc

θm

4

10 5.35.3 8

4 .5 4 .54 .5

4 .54 .5

4 .5

3 .1

4

4 53 .1

3 .1 4

4

4 32 4

44

44

4

44

(a)

(b)

(c)

(d)(e)

(f)

(g)

Note: Numbers indicatevalues of parameter M/VD

θc= θm

FIGURE 5 Evaluation of yc as a function of ym for groups of test specimens with similar

structural characteristics.



Figure 7 shows lateral force-drift ratio hysteresis loops obtained in the mentioned six

columns whose characteristics are also shown in Table 1. The insets in Fig. 7 show the

measured values of ultimate drift ratios, ym, the calculated values of yc using Eq. (11),

and the calculated values of parameter g at collapse (EH = El) obtained using Eq. (15).

The arrows in Fig. 7 correspond to the last cycle considered in the computation of EH for

the cyclic case. As can be seen in Fig. 7, different values of parameter ym were obtained

in these identical six test specimens. However, computed values for the equivalent elastic

drift yc appears to be constant for these six specimens, that is, about the same plastic work

capacity. It is important to note that that the test unit TP006 reached an ultimate drift

capacity of about three times that reached by test unit TP001.

Since they were not tested under a cyclic loading, test units TP004 and TP006 need

further explanation on how parameter yc was calculated (see Fig. 7). To be consistent

with the definition of failure given here for the cyclic case, failure in a monotonic type of

loading would correspond to a lateral strength decay of 20% at the first loading . As seen

in Fig. 7, before unloading, column units TP004 and TP006 did not undergo any strength

decay, therefore according to the given definition of failure, they did not fail, and their

results need to be evaluated properly. In fact, the testing of these column units is a

combination of both monotonic and cyclic cases.

To evaluate parameter yc for column units TP004 and TP006, it is assumed that a

cyclic loading would have at least one full incursion in the four quadrants of a plot lateral

force-displacement up to a strength decay of at least 20% at the maximum drift reached in

testing. For column unit TP004 these assumptions would lead to consider its plastic work

at the end of testing, which was equal to 51.9 kN-m. The plastic work computed for this

unit at the first quadrant was 14.4 kN-m (see Table 1), that is, the total plastic work of this

unit was about 3.6 times the plastic work of the ‘‘monotonic’’ response. Accordingly, for

the evaluation of parameter yc in column units TP004 and TP006, parameter EH was

computed considering the plastic work at the end of testing of column unit TP004. Based

on these results, it is suggested that a simplified procedure for the evaluation of EH is used

when computing Id for the monotonic case, in which EH is assumed equal to 4 times the

plastic work computed at the first quadrant. The monotonic case is further analyzed later.

0 0.02 0.04 0.06 0.08 0.1 0.12θm

θc2

0

0.003

0.006

0.009

0.012

0.015

γθc θm>

2 2θc θm=

( >1)

γθc θm<( <1)

FIGURE 6 Relationships between a dimensionless hysteretic energy at failure and ym.



Figure 8 plots values of parameter yc obtained as the ratio El / ky (see Eq. 11) for the

six test specimens as a function of observed parameter ym. Results shown in Fig. 8

indicate that the six test specimens had about the same value of equivalent elastic drift yc.

Considering Eq. (11) and that parameters k and h had similar values in this group of

specimens, these results indicate that these test units dissipated about the same hysteretic

energy at different levels of parameter ym.

According to Eq (10) for an evaluation of parameter Id, it is necessary to determine

the value of parameter yc. Results in Fig. 5 show a wide range of expected yc values.

These values depend on three main structural properties: the axial load ratio P/Ag f’c, the

–200

–150

–100

–50

0

50

100

150

200

–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1

θ

θ θ

F (kN)

F (kN) F (kN)

F (kN)

θ

–(a) Test Unit TP001 (θm = 0.034, θc = 0.051, γ = 1.5) (b) Test Unit TP002 (θ m = 0.044, θc = 0.047, γ = 1.07)

(c) Test Unit TP003 (θm = 0.058, θc = 0.043, γ = 0.74) (d) Test Unit TP004 (θm = 0.082, θc = 0.043, γ = 0.52)

(e) Test Unit TP005 (θm = 0.072, θc = 0.051, γ = 0.71) (f) Test Unit TP006 (θm = 0.087, θc = 0.047, γ = 0.54)

–200

–150

–100

–50

0

50

100

150

200

–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1

–150

–100

–50

0

50

100

150

200

–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1

–200

θ

F (kN)

–150

–100

–50

0

50

100

150

200

–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1

–200

θ

F (kN)

–150

–100

–50

0

50

100

150

200

–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1

–200

–150

–100

–50

0

50

100

150

200

–0.1 –0.08 –0.06 –0.04 –0.02 0 0.02 0.04 0.06 0.08 0.1

–200

FIGURE 7 Lateral force-drift ratio hysteresis cycles in identical columns tested by

Takemura and Kawashima [1997].



confining reinforcement and the shear-span ratio M/VD. An accurate procedure for

evaluating parameter yc as a function of the mentioned structural properties is beyond

the scope of this paper. However, the authors suggest a simple procedure to obtain an

approximation of the expected value of parameter yc for a particular case. The

procedure is based on assuming that the case under study belongs to one of the

seven groups shown in Fig. 5, after which its corresponding value of yc can be

obtained. Given the complexity of the seismic damage analysis problem addressed in

this study, in which we need to consider not only energy dissipation capacity but also

energy demand, the use of this simple procedure for evaluating yc is useful for such

damage analysis.

The above findings suggest that both static and dynamic experimental responses are

not directly comparable, since displacement history is relevant for defining the capacity

for the drift ratio of RC members. As discussed before, in such a comparison the plastic

work demands (k� �2 �2

m, see Eq. 14) need to be computed, and compared to the plastic

work capacity (k� �2c , see Eq. 11).

6. Damage Analysis of RC Members that Fail Under Monotonic Loading

Figure 9 shows an idealized force-deformation relationship for a SDOF system that fails

under monotonic loading considering an elasto-plastic curve and an incursion in the

positive quadrant with a maximum drift ratio equal to ym. The drift ratio at yielding of

this system is yy and its elastic stiffness is equal to ky. Figure 9 also shows the parameter

yc and the energy absorbed in the elastic SDOF system at collapse (with the same

frequency of the analyzed system and EH = El) when reaches the deformation yc.

Parameter EH for the SDOF system in the monotonic case is computed with the previous

recommendation that EH is equal to four times the energy dissipated by a monotonic

loading, which leads to:

EH ¼ 4 k� �y ð�m � �yÞ: (18)

Considering the collapse level (EH = El), parameter yc is calculated using Eqs. (11) and

(18), which leads to:

0

0.02

0.04

0.06

0 0.02 0.04 0.06 0.08 0.1θm

θc

TP001

TP002

TP003

TP005

TP004

TP006

FIGURE 8 Relationship between an equivalent elastic drift and ym for identical columns

tested by Takemura and Kawashima [1997].



�2c ¼ 4 �y ð�m � �yÞ: (19)

From Eq. (19), the following relationship between yc and ym is obtained:

�2c ¼

4 �2m

�m

ð1� 1

�m

Þ: (20)

Equation (20) indicates that by using the parameters ym and mm obtained in testing an RC

member that fails under monotonic loading, we can get an approximation of the energy

dissipation capacity of a similar member when it fails under cyclic loading. Some

examples are given in the following which support this hypothesis.

Let us analyze the cases of test units TP004 and TP006, which failed under monotonic

loading and were discussed in Sec. 5 of this article along with their companion test units

with similar structural characteristics but tested using different displacement histories.

Measured values of parameters ym and mm at collapse found after monotonic testing of

column units TP004 and TP006 are shown in Table 4. Results of the computation of yc with

Eq. (20) using these values are also shown in this table. As seen in Table 4, these results are

close to the previously computed values of parameter yc shown in insets of Fig. 7.

Furthermore, these computed values of parameter yc for the monotonic cases using Eq.

(20) are also close to those previously computed for the column units tested under cyclic

loading, namely, TP001, TP002, TP003, and TP005; see insets in Fig. 7.

7. Calibration of Proposed Damage Index Against Observed Evolution ofDamage in Test Specimens of the Database

A set of 21 test specimens was selected from the database chosen for this study for a

calibration of the proposed index and observed evolution of damage during testing

kθM

θθmθcθy

ryh

Eλ

FIGURE 9 Parameters involved in the evaluation of the proposed damage index for a

RC member that fails under monotonic loading.

TABLE 4 Evalution of parameter yc in monotonic tests

Column Unit ym mm yc (Eq 20) yc (Fig 7)

TP004 0.082 12 0.046 0.043

TP006 0.087 10 0.052 0.047



[Rodriguez and Padilla, 2006]. This number of test specimens was arrived at based on the

detailed description of observed evolution of damage presently available. The following

classification of damage levels was considered in this calibration:

1. Localized minor cracking or first yielding of tensile reinforcement;

2. Light cracking throughout or first yielding of compressive reinforcement;

3. Severe localized cracking or onset of localized spalling;

4. Buckling of longitudinal reinforcement.

Figure 10 shows a plot of results found for Id using Eq. (10), where EH was computed at

the drift ratio yi, corresponding to the levels of damage A–D observed during the testing

of the 21 test units. These levels of damage are identified with the symbols shown in an

inset of Fig. 10. The results shown in Fig. 10 indicate some trends on the relationships

between observed damage and parameters Id and yi/ym. These trends are summarized in

an inset of Fig. 10, which indicates that minor damage would correspond to Id < 0.1;

moderate damage would correspond to 0.1 < Id < 0.6; and severe damage would

correspond to Id > 0.6. The inset of Fig. 10 also shows the values of yi/ym that would

correspond to each of these three levels of damage.

A calibration of the Park and Ang index, IPA, against the observed evolution of

damage in the abovementioned set of 21 test units is shown in Fig. 11. As seen there, the

trend of results using the Park and Ang index is comparable to that shown in Fig. 10 for

the proposed damage index. Results in Fig.11 indicate that minor damage would corre-

spond to IPA < 0.2; moderate damage would correspond to 0.2 < IPA < 0.6; and severe

damage would correspond to IPA > 0.6.

It must be pointed out that the proposed damage index does not have the previously

mentioned disadvantages of using the Park and Ang damage index, since when um is

equal to uy the index Id is equal to zero, and Id is equal to 1 when the structure fails under

monotonic loading

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0θi/θm

Id

(A)(B)(C)(D)Id = 0.1Id = 0.6

Id Damage

Classification i / mθθ

0 to 0.1 A,B 0 to 0.3

0.1 to 0.6 C 0.3 to 0.7

0.6 to 1.0 D 0.7 to 1.0

FIGURE 10 Calibration of proposed damage index and observed damage in 21 test

specimens of the database.



8. Conclusions

This article proposes a new damage measure for the seismic damage analysis of structural

RC members. The following conclusions were obtained in this study.

1. An evaluation of results observed in laboratory tests of a set of RC column units

under different loading protocols showed the importance of displacement history

for defining the capacity for the drift ratio of RC members in a seismic event. It

follows that there is no unique capacity for the drift ratio of an RC member (or

similar ones).

2. The plastic work capacity of an RC member is nearly constant for RC members

with similar structural properties. Since at failure this work is equal to the plastic

work demand (earthquake or laboratory test demands), this leads to a definition of

a rational seismic damage measure.

3. The proposed damage measure was calibrated against experimental results

from a column database selected for this study at failure and at levels of

damage preceding failure. Results of this calibration indicate a reasonable

agreement between predicted and observed damage in the analyzed column

database.

4. There is a relationship between displacement history and the capacity for the

drift ratio of RC members, which is captured by the parameter g (or yc/ym).

This relationship can be used for defining lateral loading protocols in laboratory

tests.

5. The proposed damage index has the convenient feature that is equal to zero when

the maximum displacement is equal to the displacement at yielding, and it is equal

to 1 when the RC member fails under monotonic loading.

0.0

0.2

0.4

0.6

0.8

1.0

0.0 0.2 0.4 0.6 0.8 1.0

IPA

(A)(B)(C)(D)IPA = 0.2IPA = 0.6

θi/θm

FIGURE 11 Calibration of Park and Ang damage index, IPA, and observed damage in 21

test specimens of the database.



Acknowledgments

Professor S. Pujol from Purdue University, and four reviewers made useful suggestions

that helped to improve the manuscript. Their reviews and time are greatly appreciated.

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Damage index for seismic analysis of RC members