INTRODUCTION
Although behavior factor (R) has been an important subject in structural
engineering studies in recent years but most important studies in this
field get back to the last two decade. Among the researchers in this field.
According to this method, to compute the quantity of R by an analytical
method, it can be formulated as follows:
R = R_{A}xR_{B}xR_{C}x…xR_{N} 
(1) 
where, R_{A}, R_{B}, R_{C},...R_{N} are
parameters such as arrangement of frames, type of structural system, composition
of loads, degree of uncertainty, damping, characteristics of nonlinear
behavior in structure, characteristics of materials, ratio of building
dimensions, failure mechanism and other effective parameters. The range
of effective factors in determining R is such that it would almost be
impossible to find two buildings with identical behavior factors. In other
words, each building has its own unique features. Therefore, instead of
adding all effective factors, as mentioned in the behavior factor relation,
usually only the factors having more determinant role in the behavior
factor are studied. In this study, two main coefficients namely the structural
capacity and the force resulting from earthquake are primarily considered
and the factors that help increase the capacity and reduce the seismic
forces are determined in the following steps.
Separate researches on behavior factor, also known as Uang plasticity coefficient
method were accomplished by Uang (1991). Pandy
and Barai (1997) studied the structural sensitivity response to uncertainty.
They assumed that for every structure subjected to a given loading, the general
uncertainty is proportional to the reverse structural sensitivity response;
thus, the structural response sensitivity reduces with increasing uncertainty.
Bertero and Bertero (1999) studied uncertainty in the
seismic resistant design. In this study, they explained the main concepts of
seismic uncertainty and defined the probabilistic effect of uncertainty on structural
failure.
Wen and Song (2003) studied the reliability of structural
behavior under earthquakes. They believed that when more elements are involved
in resistance against lateral load, the probability for collapse of all elements,
at the some time, is lower than the case when le elements with equal resistance
are involved.
Husain and Tsopelas (2004a, b)
tried to determine structural uncertainty in reinforced concrete buildings.
In that, they studied r_{s} (uncertainty resistance coefficient) and
r_{v} (uncertainty variation coefficient) and their relation with the
component’s plastic rotation ductility factor (μ_{θ}).
The effects of number of stories and bays, the length of bays and story height
were studied as well. They then studied the effect of uncertainty on behavior
factor (R_{R}). Here, the effect of number of stories and bays, bays’
length, story height and also the effect of gravity loads on uncertainty coefficient
are studied. Even the effect of number of frames present at each lateral load
direction has been considered and finally, the procedure to compute uncertainty
coefficient using uncertainty resistance and uncertainty change coefficients
were studied.
REDUNDANCY
The redundancy concept has been considered by engineers, especially after
Kobe, Northridge and Turkey earthquakes, during which many buildings with
low redundancy degree were damaged. Therefore, the redundancy topic was
introduced seriously and the degree of redundancy in structural systems
was considered for seismic design.
There is some information about the useful effects of redundancy in structural
resistance, but the efficient methods measurement methods are not available
as yet. The effects of three parameters are usually considered to measure
redundancy degree, which include:
• 
Static redundancy degree of system 
• 
The ratio of probability in system failure to parts failure 
• 
Involvement of additional capacity which was not necessary for design 
Some researchers studied the effects of redundancy degree with deterministic
method; that is, use of nonlinear static analysis. There are few studies
in which probabilistic method is applied to determine the effects of redundancy
degree using structural reliability.
Seismic redundancy degree (n) for a structural system is actually the
number of critical areas (plastic hinges) in a structural system which
continue to yield until the structure exceeds the allowable limit leading
to emergency disasters like plastic displacement or complete collapse.
In engineering problems of earthquake, it is assumed that if all critical
points (plastic hinges) yield simultaneously, the structure would fail
under earthquake shaking. The redundancy degree is defined using the parallel
and serial structural system reliability theory, determining the probability
of failure in serial systems by weakest connection model and setting the
probability of failure in parallel systems through secure decay model.
Bertero and Bertero (1999) studied the effect of redundancy
and redistribution of internal forces in seismic design and stated that a part
of behavior factor is originated from redundancy degree and can not be determined
independent from overstrength and ductility. They also assumed that when the
structure can not withstand gravity loads under the effect of earthquake forces,
it would collapse. About structural resistance against displacement due to increasing
lateral load, the resistance in the first yielding point is considered and the
maximum resistance is predicted using the reliability of displacement capacity.
A structure takes advantage of the positive effects of redundancy degree
when:
• 
Change coefficient in structural demand reduces in comparison to
change coefficient in structural capacities 
• 
Addition resistance increase 
• 
Curvature capacity increases in plastic hinges 
• 
A minimum rotation capacity is ensured in all elements of structural
system 
According to much uncertainty in structural capacity and demand, one
of the methods in studying the redundancy of structural systems under
seismic loads, is to use the reliability concept. In one kind of structural
system without change in materials and configuration, the redundancy degree
factor can only influence the reliability on structural stability against
earthquake induced lateral loads and the structure behavior factor, seriously.
It should be considered that the redundancy degree is different in similar
frames. If the size of an element, its reinforcement and implementation
details change, the failure mechanism may naturally change, but even for
two completely similar frames, redundancy degree will be different for
various lateral load models.
Behavior factor used in codes, which reduces the level of elastic forces
in the design process and in its primary formulation, is defined in terms
of ductility coefficient (R_{μ}) and the additional resistance
coefficient (R_{S}). Ductility coefficient is computed considering
nonlinear response of structural system. The relationships for computing
functional ductility coefficient is formulated by some researchers by
involving the natural period of structure and its ductility capacity,
which are commonly based on nonlinear response change of a multistory
building relative to nonlinear response of a system with single degree
of freedom.
The overstrength capacity show the actual lateral resistance in comparison
to modeling resistance overstrength may be divided into two general parts.
The first part is related to the overstrength resistance modeling until
the first hinge yields in a structure and the second part is related to
formation of the first hinge until a mechanism for total failure of a
structure is developed.
In ATC19 (1995), the formulation of behavior factor
(R) is introduced. This coefficient includes an additional factor (R_{R})
used to account for the effect of redundancy degree in a structure. These effects
include probability effects and others related to structural systems geometry
either in a plan or at a point in height. Therefore, behavior factor (R) is
equal to:
R = R_{μ}.R_{S}.R_{R} 
(2) 
Some effective parameters in redundancy and structural systems reliability
are the ratio of demand to the capacity of structural systems, the kind
of failure mechanism formed, building high, the number of stories, the
length and the number of bays. This study computes the probabilistic and
deterministic effects of redundancy through obtaining two redundancy resistance
index (r_{s}) and redundancy variation index (r_{v}).
These two indexes are used in measuring the resistance reduction coefficient
from R_{R} redundancy for structural frames with twodimensional
reinforced concrete.
Redundancy indexes: Redundancy resistance index r_{s}
represent the ability of a structural system in redistributing forces
while failure and the capability of a structure in transferring the forces
of elements yielded to the elements with higher resistance. This index
is a function of static redundancy, ductility, strain hardening and the
average resistance of elements in a structural system. Second index having
probability nature is an r_{v} redundancy variations index. This
index measures the probability effect of elements resistance on structural
system resistance. It is also a function of static redundancy in a structural
system and on the other hand is a function of statistical nature in ductility
and structural elements resistance. Following variables are used in computing
above indexes:
• 
Base shear in the beginning of yielding system 
• 
Ultimate base shear 
• 
The number of local failure or the number of plastic hinges caused
during ultimate failure of structure 
• 
The access of elements curvature to ultimate curvature 

Fig. 1: 
Baseshear versus topfloor drift curve 
Redundancy resistance index: Redundancy resistance index r_{s}
are defined as the ratio of average ultimate resistance (_{u})
to yielding resistance (_{y}).
In which _{y}
is the average system resistance non redundant system.
So, in Eq. 3, both parameters _{u}
and _{y}
can be defined with respect to nonlinear static analysis curve (Fig.
1). In a method suggested for this study in studying the effects of
redundancy using nonlinear dynamic analysis with increased acceleration,
the base shear during failure and yielding is considered. In earlier studies,
this method is applied for studying the effects of overstrength. In this
study, the system failure standards that will be considered in nonlinear
static and dynamic analysis with increased acceleration are as follows:
• 
Limitations related to storey drift which according to Iran 2800
standard for buildings which period lower than 0.7 sec are limited
to 2.5% and for structures with period more than 0.7 sec are limited
to 2% 
• 
The index of structure stability which in a structure with high
ductility is limited to 0.125 and in a structure with low ductility
is limited to 0.25 
• 
The formation of failure mechanism in a structure and collapsing
structure 
• 
The access of structure failure index to a number one according to Park
and Ang (1985) criterion 
In pushover static analyses performed in this study, it is assumed that
lateral loads with reverse triangular distribution are inserted into a
structure which is proportional to Iran 2800 standard earthquake force.
In nonlinear dynamic analysis with increased acceleration, the maximum
acceleration of any record is coordinated to a primary number (here, it
is considered to be 0.02 g) and in one stage in increased to 0.02 g and
the structure is analyzed in every step until when one of the four abovementioned
criteria’s is occurred. In this stage, the analysis is stopped and
base shear is used during yielding and maximum base shear is used for
measuring r_{s} redundancy resistance index.
Redundancy variation index: The relation between resistance of
a structural system and the resistance of its composing elements is obtained
using plastic analysis of structure. In this relation, the selection of
failure mechanism is important because it can result in nonactual estimates
from redundancy variation index. For simplify computations, one sway mechanism
according to Fig. 2 is considered. This mechanism is
based on the strong column and weak beam assumption which column resistance
is at least 20% more than the resistance of beams.
The frame strength (base shear strength) for any failure mode could be
represented by the following expression:
Where:
S 
= 
Frame strength (base shear) 
n 
= 
No. of plastic hinges in the frame resulting from the particular
failure mode or collapse mechanism considered 
M_{i} 
= 
Yield moment of the structural element where plastic hinge i is
formed 
C_{i} 
= 
Coefficient with units radians length that is a function of the
plastic rotation and geometry of the structure. Equation
4 is of the form of the strength equation of a parallel system
type 
The mean value of the frame strength can be derived from the fallowing
expression:
Where:
Mi 
= 
Mean value of the strength of the structural element where plastic
hinge i is formed 
Accordingly the standard deviation of the frame strength σ_{f}
can be obtained from:

Fig. 2: 
Sway type failure mode of a generic plane frame 
Where:
ρ_{ij} 
= 
Correlation coefficient between the strengths M_{i}
and M_{j} 
σ_{Mi} 
= 
Standard deviation of the yield moment M_{i} 
ρ_{ij} 
= 
1 for i = j 
To further simplify the deviation, a regular multistory multibay frame
with the following properties is considered:
• 
The frame is composed of elements with identical normally distributed
strengths: 
• 
The correlation coefficient between the strength of any two pairs
of elements is the same: 
• 
The bays of the frame have identical spans and the stories identical
high which result in: 
Equation 5 and 6 now become:
The following relationship between the Coefficient of Variation (CV)
of the frame strength υ_{f} and the CV of the element strength
v_{e} is calculated by dividing Eq. 12 to 11:
The redundancy variation index r_{v} is defined as the ratio
between υ_{f} and v_{e}:
For a parallel system with unequally correlated elements, ρ_{e}
could be substituted with the average correlation coefficient
defined as:
Therefore, Eq. 14 can be modified using the average
correlation coefficient of the strengths of the plastic hinges as fallows:
Hence, the redundancy variation index r_{v} is a function of
the number of plastic hinges n and their average correlation coefficient
between their strengths and represents a measure of the probabilistic
effects of redundancy on the system strength, its values range between
0 and 1.
For a building structure where a single plastic hinge causes collapse (n =
1), r_{v} = 1 and the structure under consideration in non redundant.
The other extreme value r_{v} = 0 indicates an infinitely redundant
structural system and is reached either when an infinite number of plastic hinges
are required to cause collapse (practically n attains large values) or when
element strengths in a structure are uncorrelated (the average correlation coefficient
in Eq. 6 is zero).
Using Eq. 16 r_{v} can be estimated from a
pushover or dynamic analysis and for a particular value of the average
correlation coefficient of the structural member strength.
Redundancy factor R_{R}: The deterministic effect captured
by the redundancy strength index r_{s} (mainly structural indeterminacy
effects) shifts the probabilistic density function of the nonredundant
system strength towards higher values to the right, without any shape
changes.

Fig. 3: 
Effects of redundancy indices r_{s} and r_{v}
on structural system strength 
On the other hand, the probabilistic effects accordingly change
the shape of the probabilistic density curve without changing its average
value (Fig. 3).
The overall effects of redundancy on the structural strength may be completely
described by the ratio of the ultimate strength of a structural system
to the ultimate strength of nonredundant structure. Thus:
Where:
S_{u} 
= 
Structural system strength which includes all the effects
of redundancy 
S_{nr} 
= 
The same strength but for nonredundant structural system 
Assuming that the strength of a structure is distributed normally, the
characteristic or design strength of a structural system, its standard
deviation, the coefficient k is formed. Therefore, both S_{u}
and S_{nr} may be written as follows:
Where:
σ_{f} 
= 
SD of the frame strength 
σ_{f} 
= 
SD of the nonredundant frame strength 
þ_{u} 
= 
Average of the ultimate frame strengt 
þ_{nr} 
= 
Average of the nonredundant frame strength 
An expression for σ_{f} could be obtained as follows:
By virtue of _{u}
= r_{s}._{nr};
Eq. 19 results into:
Where:
r_{v} 
= 
Redundancy variation index 
r_{s} 
= 
Redundancy strength index 
v_{e} 
= 
CV of the strength of the structural system elements 
Using Eq. 19, 21 and 17
becomes:
where, v_{nr} is the CV (coefficient of variation) for nonredundant
frame strength.
A nonredundant frame structure could be modeled as a parallel system
consisting of ideal elasticbrittle elements. Such a system behaves like
a series system, where failure of one element results in the system collapse
and that the safety index of the system is equal to that of the element.
For a nonredundant system, (n = 1) v_{nr} = v_{e}. Therefore,
the redundancy factor (R_{R}) can be expressed as follows:
For a normally distributed strength not being exceeded 95% of the time,
k ranges between 1.5 and 2.5. Without any loss of generality the following
values of the CV of the element strength could be used, v_{e}
= 0.08 0.14. Whence, an average value of 0.2 , for the product kv_{e}
could be used with reasonable accuracy in evaluating the effect of r_{s}
and r_{v} on R_{R} (Fig. 4).
The expression for r_{s} =1.0 corresponds to a nonredundant
system or a system consisting of ideal elasticbrittle elements. For kv_{e}
= 0.2, probabilistic effect of redundancy on the strength of a system
could not exceed 25%. That is for a system with r_{v} = 0 and
accordingly the highest value for R_{R} = 1.25. On the other hand,
given a structure with minimal probabilistic redundancy effect, R_{R}
is proportional to r_{s}.
Case study about the effects of redundancy on twodimensional concrete
frames: In order to compute redundancy indices, 16 frame samples from
2 to 5 bay and with two, four, six and ten stories were designated. SAP2000
software and the IDARC software are used for nonlinear dynamic and nonlinear
static analysis.

Fig. 4: 
Variation of R_{R} with respect to r_{s}
and r_{v} for structural systems with kv_{e} = 0.2 

Fig. 5: 
Reinforced concrete frames with two story two bay to
ten story five bay 
For nonlinear static analysis, 16 frame samples with high ductility and
16 frame samples with low ductility are selected (Fig. 5).
The lateral load pattern applied to the structure is reverse triangular,
which is approximately in accordance with lateral force criteria of the
earthquake standard 2800 of Iran. Four different cases of design and analysis
are considered for comparison. In the first case, the bay length is 4
m and the story height is 3 m and in the second case, the story height
is increased from 3 to 4 m. In the third case, the bay length is increase
to 5 m and finally in the forth case, the gravity loading intensity is
increased to 30%. Therefore, in static nonlinear analysis, one hundred
twenty eight frames are designed with SAP2000 and then analyzed by the
IDARC. Response curves are computed in terms of displacement at the top
of structure (Δtar) with respect to base shear divided by structure
weight (C_{b}). Two values, base shear coefficient during yielding
and also maximum base shear coefficient are important over curve. The
r_{s} index is obtained by dividing maximum base shear coefficient
to the base shear coefficient when yielding.
RESULTS AND DISCUSSION
Using maximum number of plastic hinges formed in nonlinear static analysis,
one can obtain r_{v} index. As a result of having these two indices,
resistance reduction coefficient can be obtained from redundancy according
to relations in the third part. Figure 6 shows the r_{s}
changes with respect to number of bays and stories for high ductility.
Accordingly, Fig. 7 show r_{v} changes for the
first case. Finally, Fig. 8 shows R_{R} changes
for high ductility.
As it is clear from results, redundancy resistance coefficient is not
much sensitive, at both high and low ductility, to number of bays, but
it is increased by adding the number of stories. Also by increasing the
number of bays and stories, redundancy change coefficient is reduced.
As a result, redundancy factor is increased by adding the number of stories
and is not sensitive to the number of bays. Thus for computing redundancy
factor, five bays are averaged.
In order to carry out nonlinear dynamic analysis with increased acceleration
by the IDARC software, a special software is developed to do these operations
automatically. This software begins to analyze with a primary PGA value
in any stage and it continues the operation with 0.02 g increase relative
to the earlier measure, until one of the failure conditions is reached.
In this case, the value of base shear coefficient is applied for computing
r_{s} and also for computing r_{v} index. The number of
plastic hinges formed while failure is used to compute 16 frames with
high ductility and 16 frames with low ductility. Eight seismic records
are applied, equally, for both linear and nonlinear static analysis methods,
for 4 different cases. Finally, 1024 frames were analyzed with different
cases and the values of base shear coefficient while forming the first
plastic hinge.

Fig. 6: 
Variation of redundancystrength index with respect
to No. of bays and stories 

Fig. 7: 
Variation of redundancy variation index with respect
to No. of bays and stories (correlation coefficient = 0) 

Fig. 8: 
Variation of redundancy factor with respect to No. of
bays and stories (correlation coefficient = 0) 
Maximum base shear coefficient and the number of plastic
hinges when failing are used as parameters required for computing r_{s},
r_{v} and R_{R} indices. It is necessary to note that
the average values obtained from eight records is the basis for computing
above indices.
Figure 9 shows r_{s} values for frames with
different number of bays and stories for high ductility. Therefore, Fig.
10 shows r_{v} values for high ductility cases. Figure
11 shows R_{R} values in terms of the number of different
bays and stories for high ductility.
For the first case, redundancy resistance coefficient in nonlinear dynamic
method like in nonlinear static method, is not much sensitive, in both
high and low ductility, to the number of bays but with increasing the
number of stories, this coefficient would rise.

Fig. 9: 
Variation of redundancystrength index with respect
to No. of bays and stories 

Fig. 10: 
Variation of redundancyvariation index with respect
to No. of bays and stories (correlation coefficient = 0) 

Fig. 11: 
Variation of redundancy factor with respect to No. of
bays and stories (correlation coefficient = 0) 
Accordingly, redundancy
change coefficient is also reduced by increasing storey and bay numbers
and as a result, redundancy factor increases with adding the number of
stories, but is not sensitive to the number of bays. Thus for computing
redundancy factor, four bays are averaged.
Figure 12 and 13 show redundancy
factor changes, respectively for nonlinear dynamic and static methods
and for both high and low ductility. Figure 14 and
15 show the redundancy factor coefficient changes,
respectively for nonlinear dynamic and static methods in four different
conditions and high ductility. Figure 16 shows redundancy
coefficient changes for nonlinear dynamic and static method with high
ductility.

Fig. 12: 
Variation of redundancy factor with respect to No. of
stories and dynamic analysis (case 1) 

Fig. 13: 
Variation of redundancy factor with respect to No. of
story for pushover analysis (case 1) 

Fig. 14: 
Variation of redundancy factor with respect to No. of
story (high ductility and dynamic analysis) 

Fig. 15: 
Variation of redundancy factor with respect to No. of
story (high ductility and pushover analysis) 
In this study, we propose two measures, which can be used to quantify
the effects of redundancy on structural systems.
The proposed indices are the redundancy strength index r_{s}
and the variation strength index r_{v}.

Fig. 16: 
Average of redundancy factor with respect to No. of
story (low duction) 
Redundancy strength index
captures the deterministic effects of redundancy and redundancy variation
index is captures the probabilistic effects of redundancy on strength
of structural systems. These two indices seem to be better indicators
as measures of redundancy effects in structural systems than the number
of plastic hinges at failure. They depend on the number of members within
the structure, on the ductility capacity of structural members, and on
the distribution of strength and stiffness among members of the structure
.
The results relate previous finding in support on the number of plastic
hinges at failure. The redundancy variation index r_{v} decreases
(probabilistic effects of redundancy increases) as the number of bays
and floors increases. The probabilistic effects of redundancy are smaller
in frames with fewer stories (The redundancy variation index r_{v}
values are higher). Note that the r_{v} values are inversely proportional
to the numbers of plastic hinges at failure and it is expected that the
number of plastic hinges formed in frames with more stories will be larger
than in frames with less stories.
On the other hand results relate previous finding in contradiction on
the deterministic effect captured by the redundancy strength index r_{s}
(mainly effects due to structural redundancy). It can be observed that
the redundancy strength index r_{s} values and in this case the
effects of redundancy are lower for frames with fewer stories, because
r_{s} is ratio of ultimate to yield strength and its value is
affected by the number of stories and bays, span to depth ratio of beams,
story height and gravity loads.
CONCLUSION
Comparing the values obtained from analysis to compute resistance reduction
factor resulting from R_{R} redundancy, the following results
are obtained:
• 
In nonlinear dynamic analysis method with increased acceleration
(IDA), the R_{R} coefficient is increased in most conditions
with reducing ductility and show that, structures designed with low
ductility have higher R_{R} than high ductility. In the Static
Pushover Analysis method (SPO), similar to Incremental Dynamic Analysis
method (IDA), the valve of R_{R} is increased in most conditions
with lowering ductility 
• 
As observed in Incremental Dynamic analysis (IDA), R_{R}
coefficient in most conditions is increased with adding the number
of stories for elements having high ductility but in Static Pushover
Analysis method (SPO), R_{R} coefficient increases with adding
the number of stories for first and second conditions but is not much
different for third and forth conditions 
• 
Comparing the responses obtained from Static Pushover Analysis method
(SPO) with Incremental Dynamic Analysis (IDA), it is concluded that
in most conditions, R_{R} coefficients obtained from static
method are larger than dynamic method, but this difference is maximally
10%. According to results from figure, we can conclude that the results
obtained from nonlinear static method are in good agreement with results
obtained by nonlinear time history method and may be used as a reliable
method 
Finally, it should be emphasized that these results are only for frames
modeled in this study and might not hold true for all other structural
models.