# Evaluation of the Seismic Behavior of the Reinforced Concrete

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```					    Evaluation of the Seismic Behavior of the Reinforced Concrete
Structures Based on the Probabilities by Nonlinear Static Analysis

M. Javanpour
MS.C. Structural .Civil Engineering, Islamic Azad University, Dezful Branch, Iran, E-mail:
mohsen.javan1982@gmail.com

ABSTRACT: This paper presents an analytical foundation for probability-based formats for seismic design
and assessment of structures. These formats are designed to be suitable for code and guideline
implementation. The framework rests on non-linear, static seismic analysis. The formats can be used to ensure
that the structural seismic design can be expected to satisfy specified probabilistic performance objectives,
and perhaps (more novel) that it does so with a desired, guaranteed degree of confidence. Performance
objectives are presumed to be expressed as the annual probability of exceeding a structural performance level.
Structural performance levels are in turn defined as specified structural parameters (e.g., ductility, strength,
maximum drift ratio, etc.) reaching a structural limit state (e.g. onset of yield, collapse, etc.). The degree of
confidence in meeting the specified performance objective may be quantified through the upper confidence
bound on the (uncertain) probability. In order to make such statements, aleatory (random) uncertainty and
epistemic (knowledge limited) uncertainty must be distinguished.
Key words: Non-Linear, Probability, Performance, uncertainty, Seismic

1     GENERAL INSTRUCTIONS                                number 2800 offer simplified formulas to
In the recent severe earthquake, it was observed that     statistically analyze the structures’ equation which
the existing buildings possessed high levels of           of course these formulas are not held accountable to
vulnerability. Considerable point is that even            the real behavior of the structures.
modern buildings have suffered severe losses              Since the damage occurs in the plastic region, the
including heavy seismic breakdown and in some             vulnerability analysis should be done in this area.
cases overall destruction. Therefore, to reduce the       Modeling of reinforced concrete structures is a
earthquake risks in urban areas and predict the           significant step toward determination of damages
vulnerability of the existing buildings due to            resulted from earth quake. A structure model has to
probable quakes of the future, achieving an               be able to model all kinds of reinforced concrete
appropriate instruction is of the most necessary tasks    elements and present accurate analysis of their non-
of the seismic engineering.                               linear dynamic performance.
One of the commonest and most conventional                Considering complexity of the factors affecting
seismic systems applied in quake-riser regions is         structure’s behavior under earthquake condition and
reinforced concrete structure. The experience             also variety of the existing buildings, it hasn’t been
obtained from the earthquakes occurred over the past      possible to present a definite method to investigate
25 years has illustrated that even if the reinforced      the seismic vulnerability of the existing buildings so
Concrete buildings are constructed based on the           far.
Design and construction regulations of the relevant       To determine a method including the entire
country, breakdown and even destruction will              mentioned specifications is a complicated and
happen in them.                                           multidimensional task. On the one hand, capacity of
Heavy damages observed in the quake-influenced            the structural system has to be assessed according to
buildings are derived from the severe seismic             its resistance and transformability and on the other
vibrations in the non-linear limit while the              hand, through electing the probable impacts of
earthquake regulations such as Iran’s regulation          earthquake on the affected area in the form of
intensity, frequency content and duration, the non-
linear behavior of the structure should be predicted       Table 1.   Range of Each Parameter Used To Select
as much as possible and subsequently on the basis of                            Records
its results, the force or the acceleration resulting in
damages of the elements and the whole structural
system have to be calculated.                                     Soil                 B,C,D usgs classification
The main goal of this paper provide fundamental                Magnitude                        6---7.3
and novel method for calculating probability of the             distance              594488.21"---1133858.3"
vulnerability of structures using statistical                    PGA                   28.58 ---212.035 in/sec²
distribution is in addition to simply being more             Number of point             1800 --- 5961 point
efficient, more accurate performance of structures                 DT                    0.005,0.01,0.02 sec
likely to be able to offer us. (Being symmetric is one         Duration                     21.9---40 Sec
of the important characteristics of this distribution
which should be noted.)
In this course, different distributions (such as beta,
gamma, povason, Laplace, gamble etc.) were                    3   Modal Pushover Analysis (MPA)
examined but not applied due to some specific
characteristics like being unsymmetrical.                     3.1 Introduction
and it was illustrated that this distribution is very      The basics of this method were proposed by Chopra
much like the Log-normal distribution from the             & Geol [2], [3]. Applying the concept of one- degree
aspect of statistic specifications. The presented          -free structure as well as the desired seismogram
mathematics papers are suggestive of this claim, too.      allows the equivalent displacement of one-degree-
In this paper, the attempts have been made to apply        free structure to be achieved, by means of which the
the logistic distribution simultaneously with the          displacement of the main structure can be calculated.
Log-normal distribution and to compare their results.
In order to be able to achieve favorable results of a      3.2 Implication Equivalent One-Degree-Freedom
structural model also was to check the results on the             Structure
nonlinear dynamic model to calculate the likely
vulnerability of structures to be used.                    The dominant concept in the entire non-linear static
analysis methods is the equivalent one-degree-free
2   STRUCTURAL CHARACTREISTIC                              Structure(Valley [10]). Indeed, the acceleration
displacement response spectrum (ADRS) in the
The applied model was a reinforced concrete                considered mode is gained through the main
bending frame which has been designed according to         structure’s capacity curve divided by the modal
the specific plasticity principles and by means of         participation multiplied by the structure’s modal
IDARC Non-linear Software. The mentioned frame
possesses 8 storeys with 126 inch in height and also       amount at roof point ( nn ) and the normal axis of
four 197 inch openings.                                    the main structure’s capacity curve divided by the
To provide the necessary plasticity and meet the           effective modal mass ( M n* ) as well.
economical considerations, the percentage of the
columns’ armatures and that of the posts were
Objective of this process in the Capacity Spectrum
limited to 1 – 3 and 1.7 percents, respectively. The
Tables 1 contain the dimensions and kinds of the           Method is the simultaneous drawing of demand and
armatures used in the frame’s different parts.             capacity parameters in one coordinate system as well
In the current research, in order to do the Time           as determining their crossing point as the structure’s
history analysis, a series of Record were needed. So,      performance point, although in fact obtained curve
30 reformed Record were selected. To make sure of          indicates behavior of the equivalent one-degree-free
being reformed, the entire records were controlled         structure. In this bilinear curve, gradient of the first
by Seismo Signal software. All the selected records
part reveals behavior of the equivalent one-degree-
belonged to California, United States and some
controlling parameters, such as distance from the          free structure and gradient of the second part,
fault and largeness, were taken into account while         represented usually as a multiple of the first part’s
selecting records.                                         gradient, is indicative of hardening after yield.
Having this curve in hand, it is possible to determine
un-linear behavior of the one-degree-free structure.
Coordinate axes of structure curve are achieved by
an un-linear incremental static analysis. First being
changed into ADRS, they are transformed into the
un-linear behavior of equivalent one-degree-free
structure of the unit mass. It is essential to formulate
the movement equation to calculate the structure’s
response and then solve it. According to the curves
shown in Fig. 1, the movement equation of the
equivalent one-degree-free structure is presented as
(1) formula:
Fig. 1: Properties of the nth-‘mode’ inelastic SDF
         


system from the pushover curve.
Fsn ( Dn , Dn )
D  2 n Dn                      u g (t )
Ln                           (1)         In MPA method, the seismic response of each mode,
from the push of building to the target displacement
Where, Dn is the displacement response of the                           of that mode, is determined by uniform distribution
               of modal lateral force S n  m n

.Since the
equivalent   one-degree-free        structure,      Dn            and
                                                                      maximum response of building is obtained through
D n show the first and second derivatives of the                        the combination of each mode’s seismic response
selected Dn , respectively,  n is the structure’s                      with the appropriate modal combination law, the
effects of higher modes would be examined. This
frequency at the nth mode as well as time history of
method is directly applied to estimate the
Fsn
the seismic acceleration.         Can be obtained                       deformation demand (such as the roof displacement
Ln                                        and the relative displacement); however, additional
through the division of normal axis of capacity curve                   considerations are needed in order to calculate the
(base shear of the main structure) by the effective                     rotational plastic hinge and elements force. The

principal assumptions of this method are un-
modal mass ( M n* ) and u g is the ground motion.
coupling and super position of the modal responses
It is concluded form Fig .1 that stiffness of the                       in the building possessing non-rubber system, i.e.
structure is a function of displacement’s amount and                    the principal assumptions of non-linear static
direction. Through numerically solving the Eq. (1)                      method.
or direct modeling of one-degree-free structure with                    MPA method allows the seismic demand evaluation
the selected un-linear behavior, maximum                                to be achieved in two stages (Donald [5]):
displacement of the equivalent one-degree-free                              a) Execution of several one-modal pushover
structure Dn can be calculated. Thereafter, by means                            analyses for different modes so as to
of Eq. (2), maximum displacement of the main                                    determine the matching modal response in
structure’s end is obtained:                                                    the final displacement level.
b) Then, evaluation of the structure’s final
u r ,n  nr ,n Dn                                           response through combining responses of
(2)           several modes conforming to appropriate
modal combination law.
Where, r ,n is the deformation of the structure’s end
at the considered mode and Dn is maximum
4   CALCULATION OF HAZARD CURVE
displacement of the equivalent one-degree-free
PARAMETERS
structure. Indeed, if authors suppose Capacity
Spectrum Method (CSM) as a method based on
spectrum analysis, MPA method can be considered
as a method based on time history analysis. It is                         This section has dealt with the process of
notable that for both mentioned methods, by                             calculating the structure’s probability of exceeding a
application of one of the modal combination                             drift demand value by means of the logistic
methods, the results obtained can be generalized to                     distribution in order to be able to easily assess
other modes.                                                            probability of structure vulnerability. (DOE [4],
Luco[8]).
expression for the limit state frequency into
(conditional) frequencies of exceeding the limit state
for a given spectral acceleration (the adopted IM),
and composing the results by integration over all
spectral acceleration values:

H LS  .P S a  S a ,c    P S a  S a ,c / S a  y ..g S a ( y ).dy
                                           
  P  y  S a ,c  . dH Sa ( y )
            
(7)

Where S a represents the IM-based demand, S a ,c
represent the limit state capacity also expressed in
Fig .2: A typical hazard curve for spectral                      spectral acceleration terms, and  represents the
acceleration. It corresponds to a damping ratio equal            seismicity rate. We have used Eq. (8) in order to
to and a structural fundamental period of 0.95
express the PDF of spectral acceleration in terms of
seconds % 5
the increment in the spectral acceleration hazard.
Coordinate of two end points considered hazard
zone in relation Eq. (3) and the following system of                                  P  y  S a  y  y 
g Sa ( y )  lim                                  
equations can be solved.
y                y
P S a  y  y   P S a  x  dG Sa ( y ) dR Sa ( y )
lim                                              
H 1 (Sa )  0 (Sa )1                        y               y                    dy          dy        (8)
H 2 (Sa )  0 (Sa ) 2                 (3)   Where g Sa ( y ) is probability density function (PDF)
at spectral acceleration value x, and, RSa ( y ) is the
Eq. (3) on both sides of the divided                           complementary cumulative distribution function
(CCDF) at S a  y .

H 1 (Sa ) (Sa )1   H (Sa )  (Sa )1 
       
 1                       (4)
We assume that the spectral acceleration capacity is
H 2 (Sa ) (Sa )2     H 2 (Sa )  (Sa )2 
a logistic variable with the following statistical
With logarithms on both sides of the Eq. (4)                     parameters:

 H 1 (Sa )                                                        median (S a , c )   Sa ,c
ln                                     (5)                                                                                  (9)
  2
H (Sa ) 
ln(S a ,c )   S a ,c
 (Sa )1 
ln           
 (Sa ) 2 
We can observe that the first term in the integral
P[ S a  S a ,c ] can be also interpreted as the CDF of
And form there                                                   the spectral acceleration capacity at S a  y :

H 1 (Sa )                                GS a ,c (Y )  P Y  S a ,c 
H 1 (Sa )  0 (Sa )1  0                       (6)                                                                               (10)
(Sa )1
Since S a ,c is assumed to be a logistic variable, the
5   Annual Frequency of Exceeding a Limit
State, the IM-based approach                              corresponding CDF can be expressed in terms of the
standardized logistic CDF:

In this section we are going to derive the annual
frequency of exceeding a limit state, H Ls , by
following an IM-based approach. The total
probability theorem (TPT) is used to decompose the
  y            
2
1  ln y ln  S a ,c   
                       
 ln                                             2        S a ,c                   1
   S a ,c                                                           
,   .
 y  S a ,c   L  
G S a ,c ( y )  P             

    (11)                                                             2
  S a ,c         
                                      H LS   .P S a  S a ,c  
                                                               
0      (cos( )  isin( ))              (17)
          .                          .y  .dy
In order to be able to integrate Eq. (7), we use                                          y . S (1  cos( )  isin( )) 2
a ,c
integration by parts and transform the equation into
the following form:                                                         We transform the integrand into a complete square
term and take all the constant terms outside of the
  y                                 integrand:
 ln            
   S a ,c     
H LS   .P S a  S a ,c     L  
             
 .dH ( y )
 sa                               9            2

  S a ,c                                           Sa ,c 2  ,     ln  Sa ,c 
4           
                                                                 
                 
(12)
   y                                                                         1 ln y  Sa ,c 2  ln Sa ,c  
  ln                                                                       (                               )2 
    S a ,c   

 2           Sa ,c                  
  d L             .H ( y )                                                                                            
  sa
   S a ,c  
                                                                        cos( )  i sin( )  ,    cos(  )  i sin( ) 
 
                  
H LS   .P S a  S a ,c  
             
The derivative of the standard Logistic CDF can be                                         
0 . . 
1
.
 cos( )  i sin( )      .dy
(18)
calculated as:                                                                             0
y .Sa ,c 1  cos( )  i sin( )           2

   y                                                               The term inside the integral is itself the derivative
  ln             
d     S a ,c
      
 
   1 .l  ln y  ln  S a ,c                      for a standard Logistic CDF:
L                                                       (13)
dy    S a ,c           y . S a ,c 
  S a ,c             
                     H LS   .P S a  S a ,c  
                                                                                                       
 
                    
0 . . 

d
.L
 cos( )  i sin( )  .dy (19)
0
dy 1  cos( )  i sin( ) 2
After the derivative of the Logistic CDF in Eq. (13)
is substituted in Eq. (12), and the hazard term is
Noting that the integral is equal to unity, the limit
replaced by the power-law approximation from Eq.
state probability can be derived as:
(14):

(14)          H LS  P S a  S a ,c   0 . . 
                                                       (20)
 0 . Sa ,c  . cos( )  i sin( ) 
H LS   .P S a  S a ,c  
             
We can observe the power-law term outside the
0  ln y  ln  S a ,c                      (15)
 y .S .l  S
exponential is equal to the frequency of exceeding
 .y .dy
                                                  (i.e., hazard) a spectral acceleration equal to the
a ,c     a ,c                                          median spectral acceleration capacity:
If we substitute the expression for the Logistic PDF
H LS  P S a  S a ,c   H Sa (Sa ,c ).  cos( )  i sin( ) 
             
in Eq. (16) into the Eq. (15):
(21)
u
e
l (u )                                     (16)
(1  e  u ) 2
We can argue that H S a (S a ,c ) is a first-order                                      Annual Frequency of Exceeding a Limit State considering the uncertainty

approximation to the limit state probability and the                                       Beta (UH) H (Sa) Med             H (Sa) Exp (RC) Exp (UC) H (LS)

term                                    
   2 .S2a ,c / 2 ,  cos()  i sin()  is                      a     Logistic Distribution

magnifying factor that accounts for the sensitivity of                                         0.5             0.002177      0.00227 1.04571 1.00297 0.002381

the limit state probability to the randomness in the                                    Log-Normal Distribution
spectral acceleration capacity.
0.5             0.002177       0.00247 1.0395      1.0027        0.00257

6        RESULTS AND DISCUSSION

Presented in this section the results on the structure                                  Table 4. Annual Frequency of Exceeding a Limit State, the IM-based
Mpa desired and calculated probability of structural                                    approach with using Interpolation the Main Hazard Curve
vulnerability using two probability distribution
MPA Drift – Structural statistical parameters
Logistic and Log-normal pay.
K0               K              Median C            Beta C
Note: Two methods have been used to deliver
results.                                                                                 0.0015           1.0442             0.7              0.424

a. Using Power-Law Method                                                             Annual Frequency of Exceeding a Limit State
b. Using the Interpolation Main Hazard Curve                                          H (Sa)                         EXP                         H (LS)

Logistic Distribution

Table 2. Annual Frequency of Exceeding a Limit State, the IM-based                         0.000933                       1.0179                  0.00095
approach with using Power-Law Method
Log-Normal Distribution
MPA Drift – Structural statistical parameter
0.000933                       1.0397                  0.00097
K0                K            Median C               Beta C

0.0015            1.0442            0.7               0.424
Table 5. Annual Frequency of Exceeding a Limit State considering the
uncertainty, the IM-based approach with using Interpolation the Main
Hazard Curve
Annual Frequency of Exceeding a Limit State
MPA Drift – Structural statistical parameters considering the uncertainty
H (Sa)                  EXP                      H (LS)

Logistic Distribution                                                                      K0         K         Median C        Beta (RC)     Beta (UC)         n

0.002177                1.0179                  0.002216            0.0015 1.0442            0.7             0.424       0.1278          30

Log-Normal Distribution

0.002177                1.0397                     0.002263       Annual Frequency of Exceeding a Limit State considering the uncertainty

Table 3. Annual Frequency of Exceeding a Limit State considering                         Beta (UH) H (Sa) Med             H (Sa) Exp (RC) Exp (UC) H (LS)
the uncertainty, the IM-based approach with using Power-Law
Method                                                                                  Logistic Distribution

0.5              0.00933        0.00973    1.04571    1.00297 0.010205

Log-Normal Distribution
MPA Drift – Structural statistical parameters considering the uncertainty

0.5            0.00933           0.01057    1.0397     1.003       0.01102
K0       K       Median C           Beta (RC)       Beta (UC)      n

0.0015 1.0442            0.7            0.424          0.1278        30
7   CONCLUSION                                                 of Energy Facilities”, DOE-STD-1020-94, U.
S. Dept. of Energy, Washington, D. C.
This paper was a formula to calculate the probability         5.   Donald E.Grierson ,yanglin Gong , Lei Xu
of structural vulnerability (Annual Frequency of                   (2006) “ Optimal performance based seismic
Exceeding a Limit State, the IM-based) using                       design using modal pushover analysis “
Logistic probability distribution is obtained and its              journal     of    Earthquake      Engineering
results with the results of Log-normal probability                 ,Vol.10,No.1(2006) 73-96
distribution to be compared.                                  6.   FEMA 350 (2000). “Recommended seismic
According to investigations conducted by the author                design criteria for new steel moment-frame
determined that the log normal probability distribution,           buildings.” Report No. FEMA-350, SAC
the probability of the vulnerability of structures to be           Joint     Venture,     Federal     Emergency
very conservative calculation. According to the research
Management Agency, Washington, DC.
was clear that the likely vulnerability of structures using
the Log normal distribution approximately 20% of the          7.   FEMA 352 (2000). “Recommended post-
structures more vulnerable than the Logistic distribution          earthquake evaluation and repair criteria for
shows (Table. 2, 3, 4, 5).                                         welded steel moment-frame buildings.”
But according to studies carried out suggest the author to         Report No. FEMA-352, SAC Joint Venture,
calculate   the     probability   distribution     Logistic        Federal Emergency Management Agency,
vulnerability of structures to be used because in a                Washington, DC.
certain specified probability distribution Logistic           8.   Luco, N.; Cornell, C. A. (1998) “Seismic
more accurate than the Log-normal distribution                     drift demands for two SMRF structures with
offers.    The fact considers less risk, but designing             brittle connections”, Structural Engineering
structures in discussion of economic problems and                  World Wide 1998, Elsevier Science Ltd.,
desirable structural safety with regard to reasonable              Oxford, England, 1998, Paper T158-3.
risk of the most essential things. Therefore it is            9.   Valley,     M.T.,      Harris,    J.R.(1998).”
suggested to calculate the probability of structural               Application of model techniques in a
vulnerability of the Logistic distribution probability             pushover      analysis”.6th    US     National
distribution simple, symmetrical, and the likelihood               Conference on Earthquake Engineering.
and ability to accurately calculate the probability of
occurrence and the response it has caused (Bader
taking some reasonable risks) instead of Log normal
probability distribution use.

REFRENCES

1. Banon, H., Cornell, C.A., Crouse, C.B.,
Marshall, P., Nadim, F., and Younan, A. H.,
(2001) “ISO seismic design guidelines for
offshore platforms”, OMAE2001/S&R-2114,
Proceedings, 20th OMAE, Rio de Janiero,
June, 200l
2. Chopra,A.K.,Goel , R.K.(2001).” A modal
pushover analysis procedure for estimating
seismic demand for building:theory and
preliminary evaluation “.[ Report No.PEER-
2001/03].Pacific Earthquake Engineering
Research Center,University of California ,
Berkeley
3. Chopra,A.K.,Goel , R.K.(2002).” A modal
pushover analysis procedure for estimating
seismic demand for building:theory and
preliminary evaluation “.J. Earthquake
Engineering           and           Structural
Dynamics,31,561-582.
4. DOE (1994). “Natural phenomena hazards
design and evaluation criteria for Department

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