EARTHQUAKE ENGI NEERI NG AND STRUCTURAL DYNAMICS.
VOL.
10, 179193
(1982)
SEISMIC SAFETY OF REINFORCED CONCRETE MEMBERS
AND STRUCTURES
HOOSHANG BANON AND DANIELE VENEZIANO
Massachusetts Institute
of
Technology, Cambridge, Massachusetts, U.S.A.
SUMMARY
Based on cyclic load tests of largescale reinforced concrete elements and assemblages, a probabilistic model
of
member
failure is developed. The model gives the probability of survival at time
t
as a functional of damage ratio and dissipated
energy up to
t.
After extension
to
multivariate survival of several members with correlated resistance, the model is
used
to
calculate the safety
of
reinforced concrete frames subjected to given input motions. Results are in terms ofthe probability
of
local failure and of no failure anywhere in the system.
INTRODUCTION
During strong ground motions, members
of
reinforced concrete (r.c.) structures undergo cyclic deformation
and often experience permanent damage. Some members lose most of their initial stiffness and loadcarrying
capacity, thus reaching a state of effective failure.
Many factors contribute to the seismic hazard of an r.c. structure: the input is uncertain and
so
is the
structural response to a given earthquake motion. The latter includes the capacity of each member to survive
given cyclic loading and the progressive spreading of damage after initiation of failure at some critical section.
With very few and unconspicuous exceptions, the literature on seismic safety of r.c. structures has limited the
treatment of uncertainty to the input motion: one postulates that structural behaviour and failure conditions
(e.g. excessive interstorey displacement, excessive ductility) are exactly known and then performs linear
or
non
linear random vibration analysis with the ground motion modelled as a stochastic function.

By contrast,
it
is assumed here that the input motion is given (eg. as a historical record
or
a simulated time
history) and attention is shifted to uncertainty
of
behaviour and structural resistance. The purpose is not only
to look at a much neglected source of variability and hazard, but also to evaluate the damage prediction
capability of sophisticated models for inelastic dynamic analysis of r.c. structures. By considering the input
motion as deterministic it is in fact possible to use models far more detailed and realistic than those typically
employed in nonlinear random vibration.
Completion of this task has required innovative modelling of nonlinear r.c. behaviour and of probabilistic
failure of r.c. members. For both tasks, adequate data from lowcycle testing of large scale members and
assemblages was found in the literature.6
l 1
The usual way of treating lowcycle fatigue data is first to fit a
relationship between uniform stress or strain amplitude and number ofcycles to failure and then to correct this
relationship for amplitude nonuniformity." Such an approach is not applicable to r.c. members owing to
nonhomogeneity
of
the material and complex interaction between reinforcing bars and concrete. It is believed
that calculated inelastic
response
quantities such as ductility, energy dissipation, damage ratio, and cumulative
plastic deformation are more indicative of the imminence of failure than input control quantities (stress or
strain amplitude, number of cycles, etc.).
It
is also found that ductility ratio (a much used single measure ofdamage, see e.g. Reference
2)is
not by itself
a consistent damage indicator and that member reliability can be more precisely determined from
simultaneous knowledge of flexural damage ratio FDR and appropriately normalized dissipated energy
En.
These last two quantities are taken here as damage state variables. The mechanical model provides FDR and
En
as functions of time for each of the critical sections and the stochastic failure model uses these functions to
00988847/82/02017915$01.50
Received 12 September
1980
@
1982
by John Wiley
&
Sons, Ltd.
180
HOOSHANG BANON AND DANIELE VENEZIANO
calculate the evolution in time of the probability of failure of each member. The probabilistic model is based on
the notion
of
hazard function, which is extended here from ordinary onedimensional reliability problems in
time to twodimensional problems, in the (FDR,
En)
plane.
The resistances of different members in a frame are usually correlated, owing to common design, materials,
quality control and workmanship. Because probabilistic dependence is important when calculating
system
reliability, a correlated multivariate model of member failure is also developed. The model makes it simple to
calculate the probability that local failure will occur anywhere in the system. This is also the probability of
structural collapse if the system is statically determinate or moderately redundant. Numerical results are
obtained
by
subjecting realistic 4storey and 8storey frames to historical earthquakes.
THE MECHANICAL
MODEL
There is a wide variety of models to choose from for inelastic dynamic analysis of reinforced concrete frames.
Some are simple and work well for specific structural types
or
loading conditions; others, more sophisticated,
are meant to have wider applicability
or
greater accuracy.
The procedure of stochastic damage analysis developed here applies in principle to any idealization of
mechanical behaviour; in fact, it can be used with different mechanical models to compare their damage
prediction capabilities. Consider
for
example a frame subjected to a ground motion which is a known function
of time, except for a scaling intensity factor A, (e.g.
A,
=
peak ground acceleration). Interest is in the
probability
P,
of
local failure anywhere in the system and in the probability
P,
of
total collapse. When using an
inaccurate mechanical model, the functions
P,(A,)
and P,(A,) might correspond to the solid lines
in
Figure
1,
whereas the dashed lines might result from using a more sophisticated model. No matter how accurate the
model, uncertainty on the state of damage cannot be entirely suppressed because of statistical variation of
member and connection properties.
0'
Peak Ground Acceleration, Ap
Figure
1.
Hypothetical probability
of
local failure anywhere in the system
( P
,)
and
of
total collapse
( P2)
Each curve in Figure 1 may be viewed as the cumulative distribution function of system resistance in terms of
A,,
referred to a given limit state
and mechanical model.
That is, the curves in Figure 1 incorporate model
uncertainty in the calculation of damage parameters.
A
specific mechanical model is selected here which, for the present purpose, compromises best between
accuracy and complexity. Hence, the resulting procedure should be of interest in its own right as a tool of
probabilistic damage prediction.
Mechanical models discarded
as
too inaccurate are the shear beam model (masses lumped at each storey level
and interstorey treated as shear springs) and the dual component model'' which uses two beamsone elastic,
the other elastoplasticin parallel and therefore can only produce bilinear behaviour. Other models are
numerically too demanding for the present purpose. Among them, the fibre model'4* in which elements are
decomposed into many fibres, section properties are determined from hysteretic curves of steel and concrete,
and characterization of overall member behaviour requires integration along the member length. Finite
element approaches have also been found unattractive because of excessive cost of analysis.
Representation of r.c. elements as elastic beams between inelastic hinges, as in the single component model
of Gilberson,
l 6
is thought appropriate provided the hinges have realistic hysteretic and stiffnessdegradation
properties.
A
way to obtain these properties is described in detail in
and is briefly summarized here.
SEISMIC SAFETY
OF
REINFORCED CONCRETE MEMBERS AND STRUCTURES
181
Bending stiffness
Realistically
for
columns but less accurately for beams subjected to high dead loads, it is assumed that all
members deform antisymmetrically [Figure 2(a)], so that each half member can be regarded as a cantilever
beam [Figure 2(b)]. From the momentxurvature relationship of each crosssection it is then possible to match
the tip displacement of the cantilever by adjusting the flexural stiffness of the hinge. In reality, if deformation is
not exactly antisymmetric, inelastic rotation at one end of the member depends on rotation at the other end.
This dependence is neglected in the present model.
A
C
Figure 2. Idealization
of
member behaviour
Hysteretic behaviour in JEexure, shear and slippage
A modified Takeda model18*
l 9
is used to describe the hysteretic behaviour of each hinge. The original
model was developed from experimental results and includes stiffness degradation. Later studies2' have
confirmed its accuracy in predicting flexural behaviour of r.c. elements.
A
limitation
of
the Takeda model is
that it does not reproduce the observed pinching of the hysteresis loops from high shear loads
or
from slippage
of the main longitudinal reinforcement. In the variant used here, nonlinear shear and slippage effects are
modelled by adding a pair of inelastic springs at each end of the member [Figure 2(c)]. The characteristics of
the springs are found from the mechanics of shear and slippage deformation, as described in Reference 17.
The actual load deflection history of a r.c. subassemblage tested to failure' is compared in Figure
3
with the
prediction from the present model. During the experiment, the subassemblage (consisting
of
a corner joint with
two columns and a beam) was subjected to quasistatic cyclic loading. Specifically, the vertical displacement
6
at the tip
of
the beam was controlled and the required load measured. The loading scheme consisted of six
initial cycles of displacement with ductility factor
4
in the positive direction and
3
in the negative direction,
followed by cycles with higher levels of ductility until failure occurred. In this case, the columns remained
elastic throughout the test and only the beam developed inelastic action. Pinching of the hysteresis loops due
to shear and slippage is apparent.
There is good agreement between calculated and observed loaddisplacement cycles, except for the half cycle
that immediately precedes failure: clearly, in the imminence
of
failure, the behaviour
of
the critical sections no
longer corresponds to the conceptual model on which analysis is based. Overall good agreement and sudden
departure immediately before failure are features consistently observed when comparing calculated and actual
loadclisplacement histories. They will also be key features in the formulation of the probabilistic damage
model.
MEMBER FAILURE AND DAMAGE ACCUMULATION
In most of the experiments available
to
us and described in a later section, failure by near complete loss of
stiffness is a sudden phenomenon. When it happens, the analytical model ceases to be accurate (for example,
the beam in Figure
3
'fails' at the point indicated by an arrow). It was found convenient to use this fact and
define failure as the event that there is a difference of at least 20 per cent between experimental and analytical
182
HOOSHANG BANON AND DANIELE VENEZlANO
I
SPECIMEN
9
0
m
'5!O
U!O
310
B!O
 t!O
'0
l.'O
2.'0
310
U.'O
5.'0
S.'O
OISPLRCEMENT
[ I N1
(
b)
Figure 3. Experimental and analytical loaddeflection curves for specimen
9
in the experiment by ScribnerWight. The
arrow
indicates the
point
of
failure
loads at peak displacement during a cycle. Although under this definition, failure may be interpreted either as
physical nearcollapse or as inadequacy of the mathematical model, one is of course primarily interested in the
first interpretation, which is the only one to be used in the sequel. It should be noted that any reasonable
definition of failure is nearly equivalent to that given here, owing to rapid degradation of the mechanical
properties of a member after the analytical model has ceased to hold.
A key quantity in the stochastic model to be developed later is the probability
P,(t)
of failure during the next
cycle of loading, given survival to the present cycle. In general, this probability will depend on the entire
loaddeformation history up
to
the present time t, and on the load during the next cycle.
One may simplify calculation of PXt) by (1) envisioning a process ofdarnage accumulation which deteriorates
the member until such conditions are reached that catastrophic failure occurs, and
(2)
defining a damage state
associated with this process, such that the probability
P,(t)
depends on the past load4eformation history
through only the current state. It will also be assumed that the damage state has finite dimension. Doing so
dramatically reduces the complexity of the model and makes the problem of statistical inference manageable.
The next section reviews several damage parameters as candidate state variables.
SEISMIC SAFETY OF REINFORCED CONCRETE MEMBERS AND STRUCTURES
183
DAMAGE INDICATORS
Naturally, one would like state variables to be as few as possible and as informative as possible on the degree of
member degradation. Several quantities now in use as measures of inelastic damage are briefly reviewed next
and evaluated as candidate state variables.
The most popular damage indicator is perhaps rotation ductility
po,
which is the ratio between the relative
rotation of the terminal sections,
Om,,,
and the same rotation at yielding, With
Om,,
=
By+OO
and the
notation in Figure qa),
For members in antisymmetric bending,
OY
can be calculated as
0,
=
My
L/6EI,
in which
L
and El are the
member length and bending stiffness, and
MY
is the yield moment. Plastic rotation
8,
in the present single
component model coincides with inelastic hinge rotation.
(C)
Figure
4.
Definition of (a) rotation ductility,
(b)
curvature ductility and (c) damage
ratio
Another frequent measure of damage is curvature ductility
p+,
which is defined in a very similar way, except
that rotation
8
is replaced by curvature
4.
With the notation of Figure 4(b),
In the case of a bilinear momentcurvature relationship,
p4
=
1
+(M,,,MY)/qMY,
where q is the ratio
between second and initial stiffness and
M,,,
is
the maximum moment along the member.
As
is clear from the
above definition, curvature ductility only reflects the state of the most deteriorated crosssection; for this
reason it does not appear to be a consistent measure of overall member damage.
Damage ratio
(DR)
is
the ratio between the initial member stiffness
KO
and the reduced secant stiffness
K,
at
maximum displacement.2 Hence,
[see Figure 4(c)]. In this study, a modified version of damage ratio, calledjexural damage ratio
(FDR),
is
used.
FDR
is defined as the ratio of initial flexural stiffness
Kf
to the reduced secant stiffness
K,,
i.e.
K
FDR
=
_f.
Kr
(4)
184
HOOSHANG BANON AND DANIELE VENEZIANO
where
K,
=
24
E l l 2
for antisymmetric bending. With respect to DR in equation
(3),
FDR neglects elastic
shear deformation, determination of which is affected by large uncertainty. Compared with peak ductilities,
damage ratios are in general better measures of loss of integrity because they also account for strength
degradation.
It is possible that after some initial inelastic deformation, the damage and flexural damage ratios remain
essentially constant while the member continues to dissipate energy through hysteretic behaviour. Further
deterioration takes place, as indicated by the fact that catastrophic failure sometimes occurs during this phase
of response. In order to account for the phenomenon, two additional damage indicators have been defined.
One is
normalized cumulative rotation,
NCR, which is the ratio between cumulative plastic rotation of the
inelastic springs (recovery during unloading not included) and yield rotation. With reference to Figure 4(a),
cleol=
CleOI
OY
MY LJ6EI
NCR
=
where
c
denotes summation over all half cycles. The other is normalized dissipated energy,
En,
which is the
ratio between the energy dissipated by inelastic rotation at one end of the member, and half of the maximum
elastic energy stored in the member in antisymmetric bending. Hence,
f 1
fr
where
t
is time elapsed since the beginning of loading and fldz) is the rotation increment of the inelastic spring
at one end of the member during the time interval from
z
and
z
+
dz. Typically,
En
has different values for the
two halves of a member.
EXPERIMENTAL RESULTS AND DEFINITION
OF
STATE
Selection of the damage state variables has been based on physical considerations and on the result of 29 quasi
static cyclic load tests, of the type shown in Figure
3.
Laboratory data on these tests has been assembled from
six different experiments reported in References
6 1 1.
Although some results are available also from dynamic shakingtable tests,
it
is found that static tests cover a
broader range of loading patterns under bettercontrolled conditions. They also directly display the change in
stiffness and strength of the member at each deformation cycle. Although the rate of loading in a static test is
different and inertia and viscousdamping forces are absent, the stochastic model of failure inferred from these
tests should apply equally well to dynamic conditions (a similar assumption was made in Reference 2).
Special care has been taken in selecting only experiments that represent the behaviour of actual buildings
under cyclic loads; for example, all 29 tests considered here are either
on
single actualsize members or on large
scale frame subassemblages with adequate joint design.
For each test, one can calculate the damage parameters at failure, either directly from the experimental
forcedeformation time history (when available) or from results of the analytical model. In the latter case, all
that one uses from the test is the description of the structural system, the time history of imposed loads or
displacements, and the time of failure.
Comparison of damage parameters using the two procedures gives a measure of accuracy of the analytical
model: in the present case, accuracy was found to be good, especially when flexural deformation dominates
over shear deformation and slippage (for detailed comparison, see Reference
17).
Here, however, one is not
especially interested in accuracy; rather, one is concerned with the failure prediction capability of the analytical
model. For this purpose, one only needs to calculate damage parameters at failure from the mechanical model.
Should one find, for example, that failure always occurs at the same calculated value of rotation ductility, then
one could take
po
as the only damage state variable and the analytical model could be used to exactly
determine the time at which members fail.
Calculated values of five damage parameters at failure for the 29 tests are given in Table
I.
Unfortunately,
none
of
the parameters attains stable values. Variability is due in part to intrinsic differences in the resistance of
SEISMIC SAFETY
OF
REINFORCED CONCRETE MEMBERS AND STRUCTURES
185
Table
I.
Damage parameters at failure for the 29 beams
subjected to cyclic load test
Specimen
,u,
p@
FDR
En
NCR
R1 8.2
R2 8.9
R3 103
R4 13.9
R5 14.2
R6 8.9
T1 10.4
T2 14.6
T3 11.3
A4 7.8
A7 5.4
A8 5.4
A1
1
3.6
A12 3.0
H7 6.5
H9
10.5
w33 12.9
W35I 16.8
P43 14.4
s 3 13.0
s4 16.8
s 5 22.0
S6 18.8
s 7 15.1
S8 15.6
s 9 15.3
s10 16.1
s11 14.2
s12 13.9
11.4 6.8
10.5 7.8
17.1 8.6
24.0 10.6
12.1 11.0
9.7 7.4
21.8 9.0
31.2 11.4
15.0 8.5
9.0 8.7
4.2 7.3
4.2 7.3
2.4 5.1
2.1 4.0
13.3 5.6
26.9 8.3
23.4
10.0
354 11.7
21.9
11.0
11.1 10.4
12.9 13.1
15.2 14.6
13.4 14.1
9.2 13.1
10.2 12.2
11.8 14.1
20.0 144
12.7 11.4
17.0 11.3
116
107
199
98
220
182
169
63
249
257
180
145
60
54
24
78
269
451
227
158
325
415
44 1
240
245
197
197
146
140
117
113
193
70
240
187
162
36
242
282
205
168
64
60
25
84
272
417
225
142
307
356
394
249
237
245
243
157
153
different elements, and in part to inaccuracies of the analytical model (see comments on the curves of Figure 1).
Not unexpectedly, high correlation (0.98) is found between normalized cumulative rotation and normalized
energy, indicating that one of the two parameters is redundant. Quite arbitrarily, it was decided to retain
normalized dissipated energy. By itself, dissipated energy may not be a good indicator of damage: when the
imposed deformations or loads are severe, it is often found that elements fail after absorption of relatively small
amounts of energy. In this case, more informative parameters are rotation ductility and flexural damage ratio.
Correlation between these last two quantities is also high
(0.95)
and the use of flexural damage ratio is more
appealing on physical grounds.
Based on these considerations, one may select
D,
=
FDR in equation
(4)
and
D,
=
En
in equation
(6)
as the
only damage state variables. Stochastic failureprediction models for individual members and groups of
members as one may find in frames are described in the next section. These models give the probability of
failure at
t
as a function of the calculated vector state function
in the time interval from
0
to
t
(a different function for each member, in the case of groups).
Figure 5(a) displays the calculated damage trajectories in the
(D,,D,)
plane prior to failure for the
29
specimens in the sample. Trajectories start at the point
( 1,O)
and are such that
D,
and
D,
are nondecreasing
functions
of
time. The shape of each trajectory is primarily controlled by the sequence of imposed
displacements; with some erraticity due to peculiarities of member and section properties. Although the
sample is small, Figure 5(a) includes a wide variety of cases, as exemplified by the selected paths in Figure 5(b):
186
HOOSHANG BANON AND DANIELE VENEZIANO
2
400
5001
Y e
i IT
%
e
300
w
.u
Q,
0
P
200
5
=
100
0
0
3
6
9
1 2 1 5
"01
2:
F
200
0,
c
W
TJ
150
Q,
0
.

100
0
z
50i
0
c
0
Flexural Damage Ratio, D l Flexural Damage Ratio, D1
( a)
(b)
Figure
5.
Experimental damage trajectories and failure points
for
(a) all speciments in the sample, and
(b)
5
selected specimens
the trajectory marked
S9
gives the damage history of a member which failed after many cycles due to combined
flexure, shear, and slippage. Trajectory
A8
corresponds to a column which failed under high flexural and axial
loads; specimen R5 was subjected to high shear and flexure cycles, whereas specimen R4 failed in a flexure
mode after only one cycle. Specimen P43 failed primarily in flexure, after many cycles at increasing levels of
ductility.
STOCHASTIC PREDICTION OF FAILURE
Modelling of fatigue usually consists of fitting a relationship between force or displacement amplitude and
number of cycles to failure. Relationships of this type seem to work quite well for metal specimens and
structures. Not so for reinforced concrete members, primarily because of nonhomogeneity and complicated
nonlinear behaviour of the material, interaction between reinforcing bars and concrete, etc. For reinforced
concrete members, one must resort to more fundamental and unfortunately complex procedures.
Having defined
D,
and
D,
as damage state variables means that the probability of failure of a member before
time
t
depends only on the trajectory
Dl ( 4
D(z)
=
[
D,(z)l
from
0
to
t.
It also means that, given that the point
P r
D(t)
has been reached safely at time
t,
the probability of
failure at future times does not depend on the damage trajectory to
P.
This situation is reminiscent of that created by life data in actuarial and industrialreliability work, except
that a single 'damage measurement' (time) is to be replaced here with two measurements,
D,
and
D,.
The
difference is important and requires not so obvious generalization of the reliability models in time only. The
main difficulty, both in formulating the new models and in estimating their parameters,
is
possible dependence
of the failure probability not only on the present damage vector
D(t),
but
on
the entire damage history before
time
t.
This means that one cannot simply infer a joint distribution of the failure location in the
(D1, D2)
plane
using experimental failure points; rather, one must use these points
and
the experimental damage paths
to
fit
a
more general type of reliability model.
The appropriate mathematical theory for individual members has been developed by Hasofer and the
present authors in a separate paper;,, it is briefly summarized below and then extended to groups of members.
Reliability
of
single r.c. members
Reliability models with only one control parameter
T
( T
=
time, load, damage or any other suitable
quantity) may be equivalently characterized by the probability density function of
T;f(t),
or
by the socalled
SEISMIC SAFETY OF REINFORCED CONCRETE MEMBERS AND STRUCTURES
187
hazard
(or
risk) function, 1(t). With
F( t )
=
fof(z)dz being the cumulative distribution function of Tand with a
prime sign to denote differentiation, the function
A(t )
is defined,
Conversely,
Clearly, the quantity A(r)dT is the probability that failure will occur before T + ~ T, given survival at time
z.
Equivalence of the two characterizations ceases to hold in the multivariate case, for which the use of hazard
functions produces more general reliability models than joint distributions of the state variables at failure.
Added generality is in the sense
of
pathdependence of the failure probability and is exactly what one needs for
the characterization of lowcycle resistance of r.c. members.
The model fitted in Reference
22
to the data displayed in Figure 5(a) is a special case of that given next.
The hazard function in one dimension, A(t), is replaced
by
two hazard function components, 1,(D1,
D,,
dDJdD,) and A,(D,,D,,dD,/dD,), which are such that the cumulative risk in going from point A=(1,0) to
point
P
along a given path Lis given by the
line
integral along L,
p I ( D',
4,
d4/dD,)dD1
+
AZ(D1,
D,,dD,/dD,)d~,
(9)
This integral replaces the quantity
fo
A(r)dr in equation
(8).
The simplification in Reference
22
consists
of
(1)
assuming that
A,
and
1,
do not depend on dD,/dD,, and
(2)
transforming the damage variables as
0:
=
D,

1,
D;
=
bD:,
then changing to polar coordinates
(r*,
a*)
in the
(Df,
0;)
plane, and finally selecting
the constants
b
and
y
in such a way that the function
A$(r*,
a*)
in the cumulative risk,
1r(r*,cr*)dr*
+A;(r*,
a*)da*
s'
(0,O)
can be assumed identically zero. The choice
y
=
0.38
nearly linearizes the paths in the transformed plane and
minimizes inaccuracy of the simplified model. According to this simplified model and with
R*
=
r*(P),
the
survival probability at
P
along L,
QL(R*),
is
given by
r
fR'
1
Notice that this probability is still pathdependent.
A
parametric function
A:(r*,
a*)
of the type
1:(r*,
a*)
=
Cpo
eaa*(r*y
(12)
was fitted in Reference
22
to the data of Figure 5(a). Parameters of the model are
b
(for transformation from
D,
to
Or),
40,
a
and
0.
For given
b
and with
y
=
0.38,
the constants
0
and
4o
were obtained by leastsquares fitting
on Weibull paper, a distribution function of the type
F(r*)
=
1exp
{
__
f i l
(r*Y+
1 1
to the empirical CDF at the failure distances
r l
( i
=
1,
...,
29).
For paths L that are linear in the
(Df,
0;)
plane
(for
a*
=
constant), this model corresponds to assuming that, in the same plane, distance to failure has Weibull
distribution.
The remaining parameters
b
and
a
were fitted by the method of maximum likelihood and estimated to be
b
=
11
and
a
=
2.0.
It was also found that the values
b
=
1.1 and
a
=
0
have neartomaximum likelihood.
Because of this reason and because of the simplifying feature of rendering the probability of failure independent
of path, this last set of values will be used later in the analysis of system reliability. The associated hazard
I88
HOOSHANG BANON AND DANIELE VENEZIANO
function
A:(r*)
is
in which r* is distance from the origin in the plane with coordinates
0:
=
D,

1
and
0;
=
1.10
D:.38.
Lines of
equal probability of failure, 1

Qr(P),
predicted by the model are shown in Figure
6
where they are compared
with the experimental failure points.
01
I
I
I
0
5
10
15
DY
=
D,
I
Figure
6.
Contours of equal probability
of
failure and experimental failure points in the
(D:,D:)
plane
Simultaneous reliability
of
several
r.c.
members
Damage of a reinforced concrete frame subjected
to
a given ground motion depends on the simultaneous
behaviour of all its members. Realistically, one should view failure as a progressive phenomenon, which starts
at some critical section and then propagates depending on the strength and stiffness of the surviving members,
as well as on the connectivity (redundancy) of the system.
Superposed to mechanical dependence (failure of one member produces overstresses in the surviving
members), there is
probabilistic dependence
among member resistances (early failure of one element makes it
likely that all elements are weak).
Study of progressive failure allowing for both sources of dependence is now under way. The next section
avoids treatment of mechanical dependence by defining system failure as failure of any member. This model is
obviously exact for socalled series (e.g. statically determinate) systems and is appropriate for structures with
little redundancy. Probabilistic dependence among member resistances is included, according to the following
multivariate survival model.
We envision a set of
n
members (columns, beams, etc.), each of which follows a damage trajectory
Li
(i
=
1,
...,
n)
in the plane with coordinates 0: and
0:.
At time
t,
the representative point on
Li
is
Pi(t),
with
distance
rt ( t )
from the origin. Therefore, the probability that the ith member survives at time
t
is, from the
previous model,
Probabilistic dependence of the failure events originates from two sources:
(1)
(physical) correlation due to
common design, materials, and manufacturing and (2) (statistical) correlation due to uncertainty on the exact
probabilistic model, e.g. on the precise form of the function
A*
in equation
(14).
A simple and effective way to model correlation of both types is to view the functions AT(r*) and
Q(r*)
as
random,
although identical for all elements. For fixed
Q(r*),
survival of the ith member at time
t
given that
members
1,2,
...,
i

1 survive at that time is assumed to occur with the probability of equation
(19,
so
that
SEISMIC SAFETY OF REINFORCED CONCRETE MEMBERS
AND STRUCTURES
189
conditional reliability of the system,
Qs(t)] Q(r*),
is given by
n
i =
1
CQs(t)
1
Q(r*)l
=
Il
QCr3t)l
Unconditional reliability results from taking expectation of both sides
Q4t)
=
E
I?
QCr:(t)l
Q
i =
1
(16)
with respect to Q. Formally,
(17)
This last operation becomes feasible if
Q(r*)
is given parametrically with uncertainty limited to the
parameters. An obvious singleparameter choice is the mean value of the failure distances
R:,
which can be
physically associated with the average quality of design, materials, control and workmanship. One is then led
to consider a multivariate model in which the mean resistance
m*
=
E[R,*]
is the same random quantity for all
members and the conditional resistances
[RT
I
m*]
are independent, identically distributed random variables.
Let
m*
have mean
m,
and variance
a:
and
[R:
1
m*]
have mean
m*
and variance
02’.
Then it is easy to show
that the unconditional resistances have mean value
m
=
m,,
variance
o2
=
0:
+a:,
and correlation coefficients
p
=
a:/(a:
+a;)
( p
is the same for all pairs of members).
Taking the distributions of
m*
and
[R,*
I
.a*]
to be Weibull and choosing the parameters
m,
and
u2
as
m,
=
12.1, a’
=
11.2 has the appeal that the present multivariate model exactly reproduces the previous
marginal (univariate) resistance distribution in the extreme cases when
p
=
0
(independent member
resistances) and
p
=
1
(identical resistances). For intermediate values of
p,
the marginal distribution of
RT
results from convolving two Weibull distributions, but is not itself Weibull.
For any given
p,
system reliability at time
t
is
given by
n
in which
Q[r:(t)
1
m*] is
the complementary CDF of the Weibull distribution with mean
m*
and variance
a2(
1
 p)
andf,. is the Weibull density function with mean
m,
and variance
a2
p.
This expression simplifies to
for
p
=
0
and to
for
p
=
1.
NUMERICAL RESULTS
For the purpose of exemplification, two building frames previously designed according to the 1973
UBC
Code23 are chosen. They are a 4storey and an 8storey frame
of
typical dimensions and design characteristics.
When using effective (cracked) stiffnesses of beams and columns, their natural periods are found to be 086
s
and
2.0 s,
respectively. Several simplifying assumptions are made for inplane dynamic analysis, the most
important of which are:
( 1)
Only translational masses are considered, which are lumped at the nodes.
(2) The base of the frame is assumed infinitely rigid.
(3) P6
effects are taken into account by assuming constant axial load on each column, equal to that due to
(4)
Axial deformation of girders is neglected.
(5)
Only flexural deformations are considered and treated using the Takeda model.
dead load.
190
HOOSHANG BANON AND DANIELE VENEZIANO
>
0
6
30
z
w
0
w
J
4
N
20
B
Two historical earthquakes provide input motions: the 1940 El Centro and the 1952 Kern County (Olympia)
earthquakes.
The position of plastic hinges developed in the 4storey frame during the El Centro earthquake
is
shown in
Figure
7.
The number near each hinge is the probability of local failure, calculated using the model
of
the last
section [equation (13)]. Probabilities smaller than
0.01
are omitted. Figure
8
shows the damage paths on the
(Dl,
D,)
plane for the end sections of the middle beam.
As
expected, most of the inelastic energy is dissipated
during the last part
of
the motion, when the damage ratio is nearly constant.
The Kern County earthquake scaled
to
the same peak acceleration
(0.35
g)
produces no appreciable
damage; however, damage becomes noticeable if the same motion is scaled to
0.5g
(Figure
9).
end
A
end
B
r
Figure
7.
Failure probability of critical sections for the 4storey frame subjected
to
the 1940
El
Centro earthquake
( AP
=
0.35g)
Figure 8. Damage paths for member
AB
of the 4storey frame in Figure
7
subjected
to
the 1940
El
Centro earthquake
Figure 9. Failure probability of critical sections
for
the 4storey frame subjected to the 1952
Kern
County earthquake
(AP
=
0509)
Similar results for the 8storey frame subjected to the
El
Centro earthquake are displayed in Figure 10. In
this case, concentration
of
inelastic action in storeys
6
and 7 indicates important response contributions from
SEISMIC SAFETY OF REINFORCED CONCRETE MEMBERS AND STRUCTURES
191
the second mode of vibration. One may also notice that the probability of local column failures is much higher
in the 8storey frame than in the 4storey frame. This is an undesirable feature of the design of the larger
structure.
(a)
(bl
Figure 10. Failure probability of critical sections
for
the 8storey frame subjected to the 1940
El
Centro earthquake
( A,
=
0.35
9);
(a) columns, (b) girders
El
Centro
n
.O
0.6
LT
Ap=0.35g
A,
=
0.5
g
\8
Story,
El
Centro
A,,
= 0.35
g
c?
0.2
J
‘0
0.2
0.4
0.6
0.8
1.0
Correlotion
Coefficient
Figure
11.
System reliability as a function of correlation between member resistances for the 4storey frame and the 8storey frame
The effect on system reliability (seriessystem definition) of correlation between member resistances is
displayed in Figure 1
1
for the 4storey frame subjected to the El Centro and the Kern County earthquakes, the
latter scaled to
05g
peak acceleration, and for the 8storey frame in response to the El Centro record.
Reliability of the 4storey frame is very nearly the same under both input conditions. The values
for
p
=
0,l
correspond to equations (19) and (20) and typically bound reliability for intermediate levels of correlation.
Figures 11 and 12 suggest that a simple practical alternative to using equation
(18)
when
p#O,
1 is to
interpolate linearly between these extreme values of
p.
The present results are based on using flexural damage ratio and dissipated energy as a pair of damage
indicators, in place of the more popular ductility (e.g. Reference 2). Because of high correlation between
damage ratio and ductility at failure, results would not appreciably change by replacing the former with the
latter quantity; however, neither of them is by itself a reliable indicator of damage in those cases when large
amounts
of
energy are dissipated.
192
HOOSHANG BANON AND DANIELE VENEZIANO
8 
7 
6 
5 
r
&
4
I7l
3
2 
i 
In order to compare present reliability results with ductility demands, storey envelopes of rotation ductility
(/A@),
as defined in equation
(l),
are shown in Figures
12
and 13 for the two frames and the earthquake motions
considered. While it is obvious that high ductility identifies the storeys with higher probability of failure, there
seems to be little numerical consistency between local ductility demand and probability of local failure:
members with the same ductility demands may have rather different reliability, especially for earthquakes with
different characteristics.
4
3 
>
b
2
c
u)
I 

El
Centro
( A,
=
0.5
g)
I 2 3 4 5 6 1 2 3 4 5 6
Column Ductility Demand Girder Ductility Demand
Figure 12. Envelopes
of
ductility demand for the 4storey frame

Columns
___
Girders
I I I l ~ I l ~ I
Ducti l i ty
Demand
1 2 3 4 5 6 7 8
Figure
13.
Envelopes of ductility demand
for
the 8storey frame
CONCLUSIONS
Sophisticated nonlinear dynamic analysis programs allow one
to
accurately monitor the seismic behaviour of
reinforced concrete structures subjected to intense earthquake loads. The same programs are often used to
calculate such damage indicators as ductility, energy dissipation, damage ratio and cumulative plastic
deformation, but cannot equally well predict when member failure will occur.
As
demonstrated by laboratory
experiments, failure of r.c. members is a rather sudden phenomenon, which at best can be probabilistically
related to the abovementioned damage parameters.
A
probabilistic model of this type is
first
developed for
single members, then extended to several members with correlated resistances, and finally used to calculate
reliability of reinforced concrete frames under given earthquake loads. The multivariate model gives the
SEISMIC SAFETY
OF
REINFORCED CONCRETE MEMBERS AND STRUCTURES
193
probability that several members survive at time
t,
as a function of the evolution to
t
of dissipated energy and
damage ratio: for a reinforced concrete membebdissipated energy is a measure
of
cumulative damage and
damage ratio depends primarily on the size
of
the largest inelastic deformation cycle. The model can be
extended to account for other calculated damage indicators.
In the present application, system failure is assumed to occur as the
first
member fails. Work is under way to
extend the probabilistic model
for
analysis of progressive failure, i.e. to determine the random propagation of
member failures through the system, following loss of loadcarrying capacity at some critical section.
ACKNOWLEDGEMENTS
This study has been supported in part by the US. National Science Foundation under Grant
No.
ENV
7717174. Additional funding has been provided by the Italian National Research Council. Authors are grateful
to Professors J.
M.
Biggs and
H.
M.
Irvine for advice in the formulation of the mechanical model.
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