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PRACTICAL METHOD TO DETERMINE LOAD AND RESISTANCE FACTORS USED IN

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					           Practical Method To Determine Load And Resistance Factors Used In Limit State Design




   PRACTICAL METHOD TO DETERMINE LOAD AND
  RESISTANCE FACTORS USED IN LIMIT STATE DESIGN
              Yasuhiro MoriÉ, Tsuyoshi TakadaÉÉ, and Hideki IdotaÉÉÉ
       É
       Graduate School of Environmental Studies, Nagoya University, Nagoya Japan
        ÉÉ
           Graduate School of Engineering, the University of Tokyo, Tokyo Japan
   É
  ÉÉ
     Dept. of Civil Engineering and Architecture, Nagoya Institute of Technology, Nagoya
                                          Japan




                                           Abstract
   In reliability-based limit state design, the performance level of a structure can be con-
   trolled by diãerentiating load and resistance factors. In order to take fully the advantage,
   Draft Recommendation of Limit State Design by AIJ proposes a practical method for
   evaluating the factors. In the practical method, the combination of time variant loads are
   considered using Turkstra's rule and non-lognormal random variables are approximated
   to an equivalent lognormal random variables for which load and resistance factors can be
   expressed in a closed form. This paper investigates the accuracy of the practical method
   using numerical examples. It also discuss the potential improvement of the practical
   method.
   Keywords: Limit State design, Load Factors, Resistance Factors, Sensitivity Factors,
             Reliability Index, Practical Method




INTRODUCTION

The safety problem in structural engineering can be treated more rationally with probabilistic
methods. These methods provide basic tools for evaluating structural safety quantitatively.
Uncertainties in loads, material properties, and construction practice, which have been tradi-
tionally dealt with by empirical safety factors, can be taken into account explicitly and con-
sistently in probabilistic safety assessment. Based on such methods, a å rst-generation of limit
                                                                                  ed
state design codes were developed; the safety checks are associated with a speciå limit state
probability or reliability.

The design format in these codes is given with load and resistance factors or partial safety factors
with which structural design can be carried out semi-deterministically without complicated
probability analysis. In most cases, only single set of the factors is provided in the code.
Conceptually, the structural performance level can be controlled by diã    erentiating the factors
[1], which makes reliability-based design an ideal tool for performance based design. However,
intensive knowledge in the theory of probability and statistics is required to determine the
appropriate factors.


                                                 1
In order to implement fully the advantage of reliability-based design, sub-committee on Limit
State Design, Architectural Institute of Japan (AIJ) published Draft Recommendation for Limit
State Design for Building Structures in 2001 and presented three ç       ows to determine load and
                                                                                   ed
resistance factors in accordance with target reliability level [2]. In the simpliå method (Flow
I), tables of load and resistance factors are provided for typical load combinations assuming
statistical characteristics of various kinds of load events at a typical site in Japan. The target
reliability level is set to equal 1.0, 2.0, or 3.0 for the reference period of 50 years for ultimate
limit states and for the period of one year for serviceability limit states.

                                                         ed
The load and resistance factors presented in the simpliå method are intended to be used to
design a building at any site in Japan. However, there are many cases that an engineer feels
the necessity of determining the factors by himself/herself, e.g., when statistical characteristics
of load events at a construction site are diãerent from that used in the draft to determine the
factors, when he/she has up-to-date information on the statistical characteristics of load and
resistance, and when he/she wants to control the target reliability level in detail. Although
it can be carried out using probability analysis such as AFOSM (Flow III), it is complicated
for ordinary engineers. The Draft Recommendation provides simple formulae as a practical
method (Flow II) for approximately evaluating LR factors taking additional information such
as above into account.

This paper describes the practical method in detail and investigates the accuracy of the method
based on the reliability level achieved using the load and resistance factor evaluated by the
practical method. It also discusses the potential improvement of the practical method.

PRACTICAL METHOD FOR LOAD AND RESISTANCE FACTORS

Load and Resistance Factors

In general a structure is subjected to more than one time varying load processes. Design format
of limit state design with load and resistance factor can be expressed as,

                                                 X
                                                 1
                                         R
                                       ûÅ n =           i S
                                                       ç Å ni                                   (1)
                                                 i=n

in which R is the resistance of a structural component, Si (i = 1; Å Å n) is the load eã having
                                                                    Å;                  ect
the same dimension with R, û and ç are load and resistance factors, respectively, and Xn is a
                                       i
nominal value of X.

Several method have been developed to handle the combination of time varying load process
[3, 4, 5, 6]. Among them Turkstra's rule [7], is one of the most simple method, with which the
combination can be approximated as the sum of time invariant random variables. Then the
load and resistance factors can be evaluated by AFOSM.

When all the basic random variables are lognormally distributed and statistically independent
of one another, load and resistance factors can be expressed in closed forms as,


                                    p
                                       1                           ñR
                           û =                        å õ
                                             exp(ÄãR Å T Å ln R )                               (2)
                                          2
                                     1 + VR                        Rn

                                    p
                                       1                           ñSi
                           çi   =                     å õ
                                             exp(ãSi Å T Å ln Si )                              (3)
                                          2
                                     1 + VSi                       Sni


in which å is the target reliability index, ñX , õ and VX are the mean, standard deviation,
          T                                       X
and coeécient of variation (c.o.v.) of X, and ãX is the sensitivity factor of X expressed as,



                                                 2
                                 v
                                 u
                                           õ R Å ñR
                                                 ûÅ
                                 u               X
                                            ln
                       ãR    =                                                               (4)
                                 t(õ R Å ñR )2 +
                                                  n

                                    ln  ûÅ          (õ Si Å i Å Si )2
                                                      ln   ç ñ
                                                          i=1

                                 v
                                 u
                                                  õ Si Å i Å Si
                                                        ç ñ
                                 u               X
                                                   ln
                       ãSi   =                                                               (5)
                                 t(õ R Å ñR )2 +
                                                 n

                                    ln  ûÅ         (õ Si Å i Å Si )2
                                                     ln   ç ñ
                                                          i=1



in which û, ç , and ñR are unknown quantities. When only one load act on a structure, the
               i
sensitivity factors can be evaluated analytically as

                                           õ R
                                             ln        õ S
                                                         ln
                                    ãR =        ; ãS =                                       (6)
                                           õ M
                                            ln         õ M
                                                        ln


                 p
in which õ M =
          ln         (õ R )2 + (õ S )2 .
                       ln        ln


when a structure is subjected to more than one load process, Eqs.(4) and (5) have to be
evaluated iteratively. The Draft Recommendation provides simple formulae to approximate the
sensitivity factors as described in the next section.

Sensitivity Factors

When R and S1 ; Å Å Sn are all normal random variates, sensitivity factors can be expressed as
                 Å;
[1],
                                         õR         õ S
                                   ãR =     ; ãSi = i                                      (7)
             q
                                         õM         õ M

in which õ = õ2 + õ2 1 + Å Å õ2 n .
          M       R    S      Å+ S

When a structure is subjected to more than one load process, both resistance, R, and the sum of
the all load eãect, Q, are approximated to be lognormally distributed, and the their sensitivity
factors are evaluated by Eq.(6). Then the sensitivity factors for Si 's, ãSi , are evaluated by
Eq.(7), replacing õ with õ . The procedure is described in the following.
                    M        Q




                                                 v
  1) Determine the statistics of Q from those of Si 's.
                                                 u n
                                 X               uX
                                     ñSi ; õ = t
                                  n
                                                                                  õQ
                          ñQ =              Q           õ i 2;
                                                         S              VQ =                 (8)
                                   i=1                    i=1
                                                                                  ñQ


  2) Determine standard deviation of ln R, ln Q and ln M .
                                                 q     Ä        Å
                                      õ R=
                                       ln            ln 1 + VR 2 ' VR                        (9)

                                                 q     Ä        Å
                                     õ Q=
                                      ln             ln 1 + VQ 2 ' VQ                      (10)

                                           q                    q
                                 õ M =
                                  ln           õ2 R + õ2 Q '
                                                ln     ln           VR 2 + VQ 2            (11)


  3) Determine ãR and ãQ from Eq.(6)



                                                     3
                                                                    Pn                 p
  4) Determine ãSi 's from Eq.(7) and normalize them so that                 ãSi 2 =       1 Ä ãR 2 .
                                                      q
                                                                       i=1

                                                  õi
                                                   S        2
                                          ãSi =      Å 1 Ä ãR                                           (12)
                                                  õQ

      Substituing Eqs.(8)-(11) into Eqs.(6) and (12), ãR and ãSi can be evaluated by

                                        ãR = v
                                             u
                                                  ~R
                                                  õ
                                             u 2 X 2
                                                          Åu                                            (13)
                                             tõ +
                                                     n
                                               ~R      õi
                                                        S
                                                        i=1



                                        ãSi = v
                                              u
                                                   õi
                                              u 2 X 2
                                                    S
                                                           Åu                                           (14)
                                              tõ +
                                                      n
                                                ~R      õi
                                                         S
                                                        i=1

                          P
      in which õ = VR Å n ñSi . u is the safety factor to consider an error of the approx-
               ~R           i=1
      imation considered in the practical method, and u = 1.05 is recommended in the Draft
      Recommendation.


In the Draft Recommendation Eqs.(13) and (14) are expressed in terms of c.o.v. rather than
standard deviation for use in practice.

Non-lognormal Load Eãect

                                                                                 ect
In most cases R is modeled as a lognormal random variate; however, load eã such as earth-
              ect                     ect
quake load eã and wind load eã is often described by an extreme value distribution. As
load and resistance factors depend on the probability distribution of load eã   ect, it is necessary
to approximate a load eã                                           ~
                          ect, Si , as a lognormal random variate, Si , to evaluate the factors by
Eqs.(2) and (3).

                                                                      ~
In the Draft Recommendation, normalized mean value of ln Si (mean value of ln(Si =ñSi )),    ~
  É                                                       ~
                                                     ~i , VS , are determined to satisfy the following
ñln Si , standard deviation, ln õ Si , and c.o.v. of S
~                               ~ln                         i
conditions.

                                          FSi (s50 ) = GSi (s50 )                                       (15)
                                          FSi (s99 ) = GSi (s99 )                                       (16)
                                                                                   ~
in which FSi (s) and GSi (s) are probability distribution function (cdf) of Si and Si , respectively,
s50 and s99 are the value of s satisfying FSi (s) = 0:5 and FSi (s) = 0:99(0.999 if Si is described
by Type II extreme value distribution and å ï 2:5), respectively. As it is not practical to
                                                 T
solve the nonlinear simultaneous equations of Eqs.(15) and (16), the Draft Recommendation
presents the following regression formulae.

                              ñÉ Si = e0 + e1 Å Si + e2 Å Si 2 + e3 Å Si 3
                              ~ln              V         V           V                                  (17)

                                   q     Ä2 Å
                                                               2             3
                              ~ln
                              õ Si = s0 + s1 Å Si + s2 Å Si + s3 Å Si
                                               V        V         V                                     (18)
                              ~
                              VSi = exp õ Si Ä 1
                                           ~ln                                                          (19)


                                                                 ect
in which VSi is the c.o.v. of the annual maximum value of load eã before approximated to be
lognormal random variate, and ej and sj are provided in Tables 1 and 2 based on the probability
distribution of the annual maximum value. The reference period of serviceability limit state
design and ultimate limit state design are assumed to be one year and 50 years, respectively.
The annual maximum value of the non-principal loads and principal load of serviceability limit

                                                  4
       Table 1: ej in Eq.(17) and sj in Eq.(18) to approximate annual maximum value

                cdf of                            ej                                sj
       annual maximum value        e0       e1           e2     e3     s0     s1           s2     s3
               Normal             0.00     0.00         0.00   0.00   0.01   0.85        -0.49   0.14
               Type I             0.00    -0.16        -0.01   0.00   0.02   1.13        -0.67   0.20
       Type II      å î 2:5
                     T            0.00    -0.28        -0.05   0.07   0.00   1.44        -0.98   0.26
                    å > 2:5
                     T            0.00    -0.28        -0.05   0.07   0.00   1.68        -1.14   0.30



      Table 2: ej in Eq.(17) and sj in Eq.(18) to approximate 50-year maximum value

                cdf of                            ej                                sj
       annual maximum value         e0     e1            e2     e3     s0     s1           s2     s3
               Normal              0.02   1.89         -1.05   0.30   0.02   0.34        -0.32   0.11
            Lognormal             -0.01   2.34         -1.07   0.16   0.00   0.61        -0.13   0.00
               Type I              0.04   2.32         -1.43   0.43   0.05   1.59        -0.60   0.22
       Type II      å î 2:5
                     T             0.01   2.82         -2.15   0.62   0.00   1.44        -0.98   0.26
                    å > 2:5
                     T             0.01   2.82         -2.15   0.62   0.00   1.68        -1.14   0.30



              Table 3: Probabilistic model(Annual maximum and single load)
                                         mean                   c.o.v.
                                S         5.0                  0.1~1.0
                                R    å =1.0, 2.0, 3.0
                                      T                           0.2



state design are approximated as lognormal random variate using parameters in Table 1, while
50 year maximum value of the principal load of ultimate limit state design is approximated
using parameters in Table 2.

ACCURACY OF PRACTICAL METHOD

In the practical method described above,


  1) load eãects are approximately described by equivalent lognormal cdf (lognormal approx-
     imation),
  2) the sum of lognormal random variates are assumed to be a lognormal random variate for
     evaluating sensitivity factors, and
                                            ect
  3) the combination of time variant load eã is considered by Turkstra's rule.


The accuracy of these approximations depends strongly on the statistical characteristics of basic
random variables. The reliability level achieved using load and resistance factors determined by
the approximation method could be far oã from the target reliability level. Many studies have
been conducted about Turkstra's rule and it is pointed out that the rule is accurate enough in
practice when the principal load is well dominant [6, 8]. Thus, the above approximations 1)
                                                                                         ect
and 2) are investigated here. It is assumed in the following that non-principal load eã and
resistance are lognormally distributed and that u=1.0 in Eqs.(13) and (14) unless otherwise
speciå ed.



                                                   5
                  β                                      β
            1.5                                        1.5
                      β T = 1.0                                  β T = 1.0


            1.0                                        1.0



            0.5                               VS       0.5                               VS
                  0               0.5       1.0              0               0.5       1.0
                  β                                      β
            2.5                                        2.5
                      β T = 2.0                                  β T = 2.0


            2.0                                        2.0



            1.5                               VS       1.5                               VS
                  0               0.5       1.0              0               0.5       1.0
                  β                                          β
            3.5                                        3.5
                      β T = 3.0                                  β T = 3.0


            3.0                                        3.0
                      □   Normal                                  □ Normal
                      ◇   Type I                                  × Lognormal
                      ○   Type II                                 ◇ Type I
            2.5                               VS       2.5                               VS
                  0               0.5       1.0              0               0.5       1.0
                             t L = 1 year                             t L = 50 years

     Figure 1: Reliability level when single load act on a structure (S ò non-lognormal)


Equivalent Lognormal Random Variate

                                                                    ect
When a structure is subjected to a single load and the load eã is lognormally described,
the reliability level achieved using the load and resistance factors determined by the practical
method, å , agrees with the target reliability level, å . The diã
            a                                            T            erence between å and å
                                                                                       a       T
                ect
when load eã is not lognormally distributed is due to the lognormal approximation. Fig.1
                                                                                   ect
illustrates å and å as a function of the c.o.v. of the annual maximum load eã using the
              a       T
probability models shown in Table 3. Reference period, tL , is assumed to be 1 year or 50 years.
As 50-year maximum value, S50 is also described by Type II distribution with the same c.o.v.
if annual maximum value, Sa is described by Type II, only annual maximum is illustrated for
Type II distribution. å is evaluated accurately and eéciently using a probability analysis using
                         a
Fast Fourier transform [9].

Fig.1 show that å by the approximation method depends strongly on the type of cdf of load
                  a
eã and target reliability level. When Sa is described by a Type I distribution function, å
   ect                                                                                        a
is unconservative for å î 2:0 and VS ï 0:3 if tL =1 year, and the error increases with increase
                       T



                                                   6
                   Table 4: Probability model (two load act on a structure)

                                               mean              c.o.v.
                          Case1       S2        5.0             0.1~1.0
                          Case2       S2         10             0.1~1.0
                          Case3       S2         25             0.1~1.0
                                      S1        5.0             0.1, 0.4
                        All cases
                                      R    å = 1.0, 2.0, 3.0
                                            T                      0.2

                    Table 5: Probability Model (Combination of 4 Loads)

                                    r.v.       mean               c.o.v.
                                     S2         1.5                 0.4
                        CaseA        S3         1.5                 0.4
                                     S4         5.0              0.1~1.0
                                     S2         5.0                 0.4
                        CaseB        S3         5.0                 0.4
                                     S4         5.0              0.1~1.0
                                     S2         5.0                 0.4
                        CaseC        S3         5.0                 0.4
                                     S4          25              0.1~1.0
                                     S2         5.0                 0.4
                        CaseD        S3         5.0                 1.0
                                     S4          25              0.1~1.0
                                     S1         5.0                 0.1
                       All cases
                                     R     å = 1.0, 2.0, 3.0
                                            T                  0.1, 0.2, 0.4



of VS . On the other hand, if tL =50 years, the target reliability is roughly achieved. When Sa
is described by a Type I distribution function, S50 is also described by a Type I distribution
function; however, VS50 in only 0.24 even if VSa =1.0

When Sa is described by a Type II distribution function, å is a little conservative estimate
                                                              a
for å =1.0 regardless of a reference period, and decreases with increase of å . When Sa
      T                                                                        T
is a lognormal random variate and tL =50 years, å decreases with increase of å , becomes
                                                    a                              T
unconservative side å =3.0 if VS ï 0:4. When Sa is a normal random variate, å cannot be
                     T                                                            T
achieved for å î 2:0 if tL =1 year. å comes to be very unconservative estimate if VSa is large.
              T                      a
On the other hand, å is a little conservative estimate, if tL =50 years.
                     a


On the course of the analyses, it was found that the accuracy increases a little with increase of
VR . As VR increases, R=S comes to be closer to a lognormal random variate.

Load Combination

When a structure is subjected to more than one load, Eqs.(13)-(14) comes to be approximation
even if all the basic random variables are lognormally distributed. In order to investigate the
accuracy and applicability of this approximation, the combinations of two loads and four loads
are considered in this section.

Lognormal random variates

Fig.2 illustrates å when all of R, S1 , and S2 (principal load) are lognormally distributed using
                   a
the probability model presented on Table 4. For Case 1, in which ñS2 = ñS1 , where ñX is the


                                                 7
                  β                                             β
            1.5                                               1.5
                      β T = 1.0         ×   Case1                       β T = 1.0         ×   Case1
                                        □   Case2                                         □   Case2
                                        ○   Case3                                         ○   Case3

            1.0                                               1.0



            0.5                                     VS        0.5                                     VS
                  0               0.5           1.0                 0               0.5           1.0
                  β                                             β
            2.5                                               2.5
                      β T = 2.0                                         β T = 2.0


            2.0                                               2.0



            1.5                                     VS        1.5                                     VS
                  0               0.5           1.0                 0               0.5           1.0
                  β                                                 β
            3.5                                               3.5
                      β T = 3.0                                         β T = 3.0


            3.0                                               3.0



            2.5                                     VS        2.5                                     VS
                  0               0.5           1.0                 0               0.5           1.0
                             VS1 = 0.1                                         VS1 = 0.4

                      Figure 2: å under combination of twoload(S ò lognormal)
                                 a



mean of X, and the principal load is weakly dominant, å is unconservative. å is very close
                                                       a                    a
to å if VS1 =0.1. On the other hand, å come to be unconservative when VS1 =0.4 and S2 is
    T                                  a
weakly dominant. On the course of analyses, it was found that such trends are fairly lightly
dependent on VR .

The dependency of å on the \dominance of the principal load" under the combination of four
                    a
load S = S1 + S2 + S3 + S4 is illustrated in Fig.3 using probability model presented in Table 5.

In general, the principal load is dominant if the mean value and/or c.o.v. of the principal
       ect                                                            ed
load eã is larger than the other loads. The dominance can be quantiå by the ratio of the
                                           ect
standard deviation of the principal load eã to that of the combination of the others, rõ.

                                            rõ = p
                                                            õ4
                                                             S
                                                                                                           (20)
                                                      õ 12 + õ 22 + õ 32
                                                       S      S      S



It is assumed that all load eã are lognormally distributed, and VR = 0:2. Safety factor u in
                              ect

                                                          8
                  β                                             β
            1.5                                               1.5
                      β T = 1.0            CaseA                        β T = 1 .0           CaseA
                                           CaseB                                             CaseB
                                           CaseC                                             CaseC
                                         × CaseD                                           × CaseD
            1.0                                               1.0



            0.5                                      rσ       0.5                                      rσ
                  0                0.5         1.0                  0                0.5         1.0
                  β                                             β
            2.5                                               2.5
                      β T = 2 .0                                        β T = 2 .0



            2.0                                               2.0



            1.5                                      rσ       1.5                                      rσ
                  0                0.5         1.0                  0                0.5         1.0
                  β                                                 β
            3.5                                               3.5
                      β T = 3 .0                                        β T = 3.0



            3.0                                               3.0



            2.5                                      rσ       2.5                                      rσ
                  0                0.5         1.0                  0                0.5         1.0

                         VR=0.2, u=1.0                                     VR=0.2, u=1.05

                       Figure 3: å under Combination of Four Loads (VR = 0:2)
                                  a



Eqs.(13) and (14) is set to be 1.0 or 1.05. In Cases A and C, the mean value of the principal
load eã ect, ñS4 , is relatively large, and accordingly the principal load is strongly dominant. In
Case B, ñS4 is equal to the mean values of the other loads. In Case D, VS3 is increased from
that of Case C to 1.0, and accordingly the dominance of the principal load is weakened.

Except for Case B, å for all the cases takes similar values when principal load is dominant
                     a
                                                                                 es
(rõ ï 3). Although å is a little unconservative when u = 1:0, it roughly satiså the target
                      a
level using u = 1:05. For Case B, å takes similar values to the other cases when å ï 2:0.
                                    a                                                T
However, the accuracy decreases to the unconservative side with increase of rõ when å = 1.0.
                                                                                     T


Fig.4 illustrates å assuming VR =0.1 or 0.4 and using u = 1:05. Similar to the case when
                   a
VR = 0:2, the target reliability level can be roughly achieved when rõ ï 4 using u = 1:05.

Non-lognormal load eãects

Fig.5 illustrate å when a structure is subjected to the combination of two loads and when S2
                  a



                                                          9
                  β                                              β
            1.5                                                1.5
                      β T = 1.0            CaseA                         β T = 1 .0           CaseA
                                           CaseB                                              CaseB
                                           CaseC                                              CaseC
                                         × CaseD                                            × CaseD
            1.0                                                1.0



            0.5                                      rσ        0.5                                      rσ
                  0                0.5         1.0                   0                0.5         1.0
                  β                                              β
            2.5                                                2.5
                      β T = 2 .0                                         β T = 2 .0



            2.0                                                2.0



            1.5                                      rσ        1.5                                      rσ
                  0                0.5         1.0                   0                0.5         1.0
                  β                                                  β
            3.5                                                3.5
                      β T = 3 .0                                         β T = 3.0



            3.0                                                3.0



            2.5                                      rσ        2.5                                      rσ
                  0                0.5         1.0                   0                0.5         1.0

                         VR=0.1, u=1.05                                     VR=0.4, u=1.05

                  Figure 4: å under Combination of Four Loads (VR =0.1,0.4)
                             a



is described by Type I or Type II extreme value distribution using the same probability models
used for Fig.2 except that VS1 = 0:1. å depends strongly on the type of probability distribution
                                        a
and target reliability level. The error is similar to that due to lognormal approximation shown
in Fig.1, implying the necessity of more accurate lognormal approximation.

REFINEMENT OF LOGNORMAL APPROXIMATION

The possibility of reåning the practical method presented in the Draft Recommendation of
Limit State Design is described in this section.

Lognormal Approximation Considering åT


For lognormal approximation, two conditions are necessary to determine two parameters of
lognormal distribution function. In the Draft Recommendation Eqs.(15) and (16) are used for
simple approximation. However, numerical examples in the previous section implied that å     T
should be considered in the approximation. One of the possibility is to use Eq.(21) in stead of


                                                          10
                  β                                             β
            1.5                                               1.5
                      β T = 1.0         ×   Case1                       β T = 1.0
                                        □   Case2
                                        ○   Case3

            1.0                                               1.0
                                                                                          ×   Case1
                                                                                          □   Case2
                                                                                          ○   Case3
            0.5                                     VS        0.5                                     VS
                  0               0.5           1.0                 0               0.5           1.0
                  β                                             β
            2.5                                               2.5
                      β T = 2.0                                         β T = 2.0


            2.0                                               2.0



            1.5                                     VS        1.5                                     VS
                  0               0.5           1.0                 0               0.5           1.0
                  β                                                 β
            3.5                                               3.5
                      β T = 3.0                                         β T = 3.0


            3.0                                               3.0



            2.5                                     VS        2.5                                     VS
                  0               0.5           1.0                 0               0.5           1.0

                          S2~Type I                                          S2~Type II

             Figure 5: å under Combination of Two Loads (S2 - non-lognormal)
                        a



Eq.(16) so that å agrees with å when a structure is subjected to a single load.
                 a             T

                                                      å =å
                                                       a  T                                                (21)


                                                    ~
The c.o.v. of the approximated random variable, VS , å = 1:0; 2:0; 3:0 is illustrated in Figs.6
                                                          T
and 7 for tL = 1 year and 50 years, respectively. It is assumed in the illustration that VR =0.1,
0.2, 0.3,or 0.4. The solid lines in the å                                          ~
                                           gures are the regression formula for VS as a cubic
                  ~
function of 1=å . VS depends strongly on not only the type of probability distribution of S but
               T
also å . However, it depends fairly lightly on VR except for a couple of cases.
      T


Fig.8 illustrates å using lognormal approximation described in this section using the probability
                   a
models presented in Table 4. The accuracy of å is improved substantially from Fig.5 and
                                                  a
compatible with that when all the basic random variables are lognormally distributed (see
Fig.2).



                                                         11
             ~                                       ~
             VS                                      VS
                   β T = 1.0                                     Regress.
            1.0                                      1.0     ○   VR=0.1
                                                             ◇   VR=0.2
                                                             □   VR=0.3
                                                             ×   VR=0.4

            0.5                       Regress.       0.5
                                  ○   VR=0.1
                                  ◇   VR=0.2
                                  □   VR=0.3
                                  ×   VR=0.4
                                              VS                            β T = 1 .0 V
              0                                         0                               S
               0                0.5        1.0           0           0.5           1.0

                   β T = 2 .0
            1.0                                      1.0


            0.5                                      0.5


                                              VS                            β T = 2 .0 V
              0                                       0                                 S
               0                0.5        1.0          0            0.5           1.0
             ~                                       ~
             VS                                      VS
                   β T = 3 .0
            1.0                                      1.0


            0.5                                      0.5


                                              VS                            β T = 3 .0 V
              0                                         0                               S
               0                0.5        1.0           0           0.5           1.0

                      S~Normal                               S~Type II
                                           ~
                                 Figure 6: VS Considering å (tL = 1 year)
                                                           T



CONCLUSIONS

This paper investigate the accuracy and applicability of the practical method for evaluating
load and resistance factors introduced in the Draft Recommendation of Limit State Design
for Building Structures by the subcommittee of Limit State Design of AIJ. For ultimate limit
state design the reliability level achieved using the load and resistance factors evaluated by the
                                    es
practical method generally satiså the target level. However, it would be very unconservative
for serviceability limit state design with tL = 1 year, and when å ô 1:0, principal load is
                                                                       T
described by normal distribution or Type I distribution, and its c.o.v. is large. It would also be
very unconservative when the principal load is weakly dominant. As the accuracy of Turkstra's
rule is also in question when the principal load is weakly dominant, special attention should be
made to apply the practical method to such cases.

This paper also presented that the accuracy of the practical method could be improved by
reåning the conditions of lognormal approximation. As it is impractical to solve nonlinear
simultaneous equations during design process, simpler approximation formulae needs to be



                                                   12
            ~                                          ~
            VS                                         VS
                       Regress.                               β T = 1.0
                   ○   VR=0.1
                   ◇   VR=0.2                      0.3
            0.4    □
                   ×
                       VR=0.3
                       VR=0.4
                                                   0.2
                                                                                 Regress.
            0.2                                                              ○   VR=0.1
                                                   0.1                       ◇   VR=0.2
                                                                             □   VR=0.3
                                  β T = 1.0 V                                ×   VR=0.4
                                                                                          VS
             0                               S          0
               0          0.5            1.0              0                0.5         1.0
            ~                                          ~
            VS                                         VS
                                                              β T = 2 .0
                                                   0.3
            0.4
                                                   0.2
            0.2
                                                   0.1
                                  β T = 2 .0 V                                            VS
             0                                S         0
              0           0.5            1.0              0                0.5         1.0
            ~                                          ~
            VS                                         VS
                                                              β T = 3 .0
                                                   0.3
            0.4
                                                   0.2
            0.2
                                                   0.1
                                  β T = 3 .0 V                                            VS
              0                               S         0
               0          0.5            1.0             0                 0.5         1.0

                   S~Lognormal                                    S~Type I
                                     ~
                           Figure 7: VS Considering å (tL = 50 years)
                                                     T



developed for use in practice.

ACKNOWLEDGMENT

                                                          c
Support of this research through Grant-in-Aid for Scientiå Research (C) from the Ministry
of Education, Science, Sports, and Culture and Japan Society for the Promotion of Science is
gratefully acknowledged.


REFERENCES

[1] Melchers RE. Structural reliability; analysis and prediction - Second edition. West Sussex,
    UK: JohnWiley and Sons, 1999
[2] Draft Recommendation of Limit State Design for Building Structures, 2nd Edition. Sub-
    committee on Limit State Design, AIJ, Japan. 2001.12. (in Japanese)
[3] Wen YK. Statistical combination of extreme loads. J. Str. Div., ASCE Vol.103, No.5,
    pp.1079-1093, 1977.5

                                                  13
                  β                                              β
            1.5                                                1.5
                      β T = 1.0         ×   Case1                        β T = 1.0         ×   Case1
                                        □   Case2                                          □   Case2
                                        ○   Case3                                          ○   Case3

            1.0                                                1.0



            0.5                                      VS        0.5                                     VS
                  0               0.5           1.0                  0               0.5           1.0
                  β                                              β
            2.5                                                2.5
                      β T = 2.0                                          β T = 2.0


            2.0                                                2.0



            1.5                                      VS        1.5                                     VS
                  0               0.5           1.0                  0               0.5           1.0
                  β                                                  β
            3.5                                                3.5
                      β T = 3.0                                          β T = 3.0


            3.0                                                3.0



            2.5                                      VS        2.5                                     VS
                  0               0.5           10                   0               0.5           1.0

                          S2~Type I                                           S2~Type II

                         Figure 8: å of Reå
                                    a      ned Method(S2 ò non-lognormal)


[4] Larrabee RD, Cornell CA. Combination of various load processes. J. Str. Div., ASCE,
    Vol.107, ST1, pp.223-239, 1981
[5] Kohno M, Sakamoto J. Exceedence probability of stochastic load combinations. Proc. ICASP
    6. Mexico: Mexico City, pp.622-629, 1991.
[6] Mori, Y. and K. Murai. Load and resistance factors taking combined load processes into
    account. Applications of Statistics and Probability, Proc. ICASP8, 1999.12, pp.973-980.
[7] Trukstra CJ, Theory of structural design decision, Study No.2. Solid Mechanics Division.
    Univ of Waterloo, Waterloo, Ontario, Canada. 1970
[8] Pearce HT, Wen YK. Stochastic combination of load eã       ects. J. Str. Div., ASCE, Vol.110,
    No.7, pp.1613-1629, 1984.7
[9] Sakamoto J, Mori Y, Sekioka T. Probability analysis method using Fast Fourier transform
    and its application. J. Structural Safety, Vol.19, pp.21-36, 1997.




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