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Discussion on: First-order third-moment reliability method
Structural safety
ELSEVIER
Structural
Safety
17 (1996)
255-257
Discussion on:
First-order third-moment reliability method M. Tich$ Structural
Safety, 16 (1994) 189-200
Z. Sadovskf, D. PhleS Institute of Construction
and Architecture
of
the Slovak Academy of Sciences, Bratislalla,
Slovakia
The paper is appealing and the topic worth of further study. According to our opinion the distributions of resistances in buckling and post-buckling problems may also possess skewnesses which completely differ from those implied by the assumptions adopted in various codes. For example, resistance of an elastic axially compressed column has the values of the coefficient of skewness in the range +0.5 depending mainly on the distribution of initial deflection [6]. Let us firstly show some details. Expanding g( .> in a Taylor series at an arbitrary point formula for the mean of Z can be found x0 = (xol, x02,. *. ~~~1, an approximate PZ
=
gCxO)
+
Cgi(xO)(PLxi
-xOi),
(21)
while the first-order terms of ~2, ~yz preserve the form of those in (7), (81, respectively. For the transformed reliability margin Z”, the quasi-mean related to the design point x0 = xd is
QP> =
- Cg,U(u(Xd))ui(Xd)*
Standardization zu =
(22)
of Z” to
ZU-&
(23)
6 obviously
yields
& = 0, and the P,-fractile
a;= 1, of Z’,
Ly;=ffg denoted
by Zi,
(24) as
(25) Elsevier Science B.V. SSDZ 0167-4730(95)00014-3
256
Discussion
/Structural
Safety
I7 (1996)
255-257
Table 3 Example 1: Probability of failure Method
FORM
SORM
Monte Carlo
FOTM
Pf
0.0162
0.0113
0.0112
0.0108
The author’s approach admits two levels of approximations: (i) the mean, standard deviation and coefficient of skewness of Z” are substituted by their first-order terms in Taylor’s expansion at the design point - the quasi-parameters, and (ii) for gU a three-parameter distribution (recommended log-normal) is chosen and adjusted to (24). We note that the main source of error imposed by (i) in the cases of non-linear limit state functions may not be the truncation of higher order terms but rather the utilization of the Taylor series itself. Because of the limited convergence radius of the series, the tails of expanded functions are not properly approximated and systematic errors may occur. Numerical investigations support this conclusion, cf. [4, p.591. Adopting (i) it follows that
Q/-G PHL= E
(26)
and by the distribution (ii) Qc$ and -pHL failure. We consider a few examples. Example
determine uniquely the FOTM probability of
1 (of the paper). Probably because of an insufficient sample size the value psi”’ = 0.0086
is unprecise. Recalculations by a Monte Carlo simulation as well as FORM and SORM show an excellent agreement with FOTM, see Table 3. 2. The fundamental case of reliability margin Z = R - S with normally distributed action effects S and three-parameter log-normal resistance R is considered. The limit state function in the standardized space is taken in the form
Example
parametrized by the reliability index /? = (puR- ~~$1,’/u; + V: and the ratio uR/us. Since gU is function, only an error due to assumption (ii) is committed when applying FOTM. The probabilities of failure presented in Table 4 show good agreement of FOTM with FORM and SORM. a linear
Table 4 Example 2: Probabilities of failure at p = 3.09 and varying cyR and a,/~,~ cyR Parameter 0.5 - 0.5 OR iuS 0.7 1.0 3.0 0.7 Method
FORM SORM FOTM
0.000739 0.000672 0.000607
0.000496 0.000447 0.000357
0.000053 0.000048 0.000030
0.00146 0.00163 0.00151
1.0
3.0
0.00208 0.00224 0.00205
0.00412 0.00416 0.00406
Discussion
/Structural
Safety
I7 (1996)
255-257
257
Unfortunately, from the viewpoint of codification, no (formal) separation of parameters in a reliability condition or/and a simple relationship to the reliability index p appears to issue from the FOTM method. Recently, this problem was successfully treated by Mrazik [7] or by an asymptotic analysis of reliability margin [8]. As an example of the nonlinear problem, we studied the reliability margin Z = R - S taking R as the resistance of an axially compressed elastic column [6] and normally distributed S. Several alternatives were calculated: (1) S = R0.05,(2) S = RO,OO1, (3) S0,95= R0,05and uR/gs = 1, 2 or 4. Comparison of the FOTM method with a Monte Carlo simulation showed that the probabilities of failure vary with slenderness and distribution of initial deflection from very close values to those differing by one order of magnitude. We congratulate the author to his contribution to the invariance concept and express our belief that it will enhance introductions of higher levels of reliability methods in new structural design codes and standards. References [6] Z. Sadovsky and D. P&S, Buckling resistance of members under compression and European standards unification, IXth Int. Con& Metal Structures, Krakow, 26-30 June 1995. [7] A. Mrazik, Separation of random variables in reliability condition with asymmetric distribution of probability, submitted for publication. [8] Z. Sadovsky, Asymptotic analysis of safety margin with normal action effects and log-normal resistance, in preparation.
Author’s
reply
I would like to express my appreciation of the Writers’ effort leading to further clarifying the FOTM Method, and thanks for helpful positive criticisms. The Writers’ comments on the limitations of the Taylor series application are certainly correct. It must be always kept in mind that the various applications of the HL-beta give only approximate information on the failure probabilities though the degree of agreement with the real failure probability is good. There are obviously many possibilities of developing the FOTM Method which might be of interest in the solution of advanced reliability problems. It is rather amazing that the FOTM Method had not been defined earlier, as it is a logical expansion of the original Hasofer’s and Lind’s idea. The Readers might be interested by the fact that the method is a result of “playing computer”. I only wanted to know what would happen if the non-normality of basic variables would be expressed by introducing the coefficients of skewness into the calculation model. I did not expect the phenomenon of the quasi-skewness invariance at all, and it took a lot of time to find the proof for this obvious feature. Erratum. Readers should kindly correct the following three-parameter log-normal probability distribution:
u,=&z+b,‘/” + la -f3p3+ a
1).
equation in the Appendix
on the
(A-4)
ELSEVIER
Structural
Safety
17 (1996)
255-257
Discussion on:
First-order third-moment reliability method M. Tich$ Structural
Safety, 16 (1994) 189-200
Z. Sadovskf, D. PhleS Institute of Construction
and Architecture
of
the Slovak Academy of Sciences, Bratislalla,
Slovakia
The paper is appealing and the topic worth of further study. According to our opinion the distributions of resistances in buckling and post-buckling problems may also possess skewnesses which completely differ from those implied by the assumptions adopted in various codes. For example, resistance of an elastic axially compressed column has the values of the coefficient of skewness in the range +0.5 depending mainly on the distribution of initial deflection [6]. Let us firstly show some details. Expanding g( .> in a Taylor series at an arbitrary point formula for the mean of Z can be found x0 = (xol, x02,. *. ~~~1, an approximate PZ
=
gCxO)
+
Cgi(xO)(PLxi
-xOi),
(21)
while the first-order terms of ~2, ~yz preserve the form of those in (7), (81, respectively. For the transformed reliability margin Z”, the quasi-mean related to the design point x0 = xd is
QP> =
- Cg,U(u(Xd))ui(Xd)*
Standardization zu =
(22)
of Z” to
ZU-&
(23)
6 obviously
yields
& = 0, and the P,-fractile
a;= 1, of Z’,
Ly;=ffg denoted
by Zi,
(24) as
(25) Elsevier Science B.V. SSDZ 0167-4730(95)00014-3
256
Discussion
/Structural
Safety
I7 (1996)
255-257
Table 3 Example 1: Probability of failure Method
FORM
SORM
Monte Carlo
FOTM
Pf
0.0162
0.0113
0.0112
0.0108
The author’s approach admits two levels of approximations: (i) the mean, standard deviation and coefficient of skewness of Z” are substituted by their first-order terms in Taylor’s expansion at the design point - the quasi-parameters, and (ii) for gU a three-parameter distribution (recommended log-normal) is chosen and adjusted to (24). We note that the main source of error imposed by (i) in the cases of non-linear limit state functions may not be the truncation of higher order terms but rather the utilization of the Taylor series itself. Because of the limited convergence radius of the series, the tails of expanded functions are not properly approximated and systematic errors may occur. Numerical investigations support this conclusion, cf. [4, p.591. Adopting (i) it follows that
Q/-G PHL= E
(26)
and by the distribution (ii) Qc$ and -pHL failure. We consider a few examples. Example
determine uniquely the FOTM probability of
1 (of the paper). Probably because of an insufficient sample size the value psi”’ = 0.0086
is unprecise. Recalculations by a Monte Carlo simulation as well as FORM and SORM show an excellent agreement with FOTM, see Table 3. 2. The fundamental case of reliability margin Z = R - S with normally distributed action effects S and three-parameter log-normal resistance R is considered. The limit state function in the standardized space is taken in the form
Example
parametrized by the reliability index /? = (puR- ~~$1,’/u; + V: and the ratio uR/us. Since gU is function, only an error due to assumption (ii) is committed when applying FOTM. The probabilities of failure presented in Table 4 show good agreement of FOTM with FORM and SORM. a linear
Table 4 Example 2: Probabilities of failure at p = 3.09 and varying cyR and a,/~,~ cyR Parameter 0.5 - 0.5 OR iuS 0.7 1.0 3.0 0.7 Method
FORM SORM FOTM
0.000739 0.000672 0.000607
0.000496 0.000447 0.000357
0.000053 0.000048 0.000030
0.00146 0.00163 0.00151
1.0
3.0
0.00208 0.00224 0.00205
0.00412 0.00416 0.00406
Discussion
/Structural
Safety
I7 (1996)
255-257
257
Unfortunately, from the viewpoint of codification, no (formal) separation of parameters in a reliability condition or/and a simple relationship to the reliability index p appears to issue from the FOTM method. Recently, this problem was successfully treated by Mrazik [7] or by an asymptotic analysis of reliability margin [8]. As an example of the nonlinear problem, we studied the reliability margin Z = R - S taking R as the resistance of an axially compressed elastic column [6] and normally distributed S. Several alternatives were calculated: (1) S = R0.05,(2) S = RO,OO1, (3) S0,95= R0,05and uR/gs = 1, 2 or 4. Comparison of the FOTM method with a Monte Carlo simulation showed that the probabilities of failure vary with slenderness and distribution of initial deflection from very close values to those differing by one order of magnitude. We congratulate the author to his contribution to the invariance concept and express our belief that it will enhance introductions of higher levels of reliability methods in new structural design codes and standards. References [6] Z. Sadovsky and D. P&S, Buckling resistance of members under compression and European standards unification, IXth Int. Con& Metal Structures, Krakow, 26-30 June 1995. [7] A. Mrazik, Separation of random variables in reliability condition with asymmetric distribution of probability, submitted for publication. [8] Z. Sadovsky, Asymptotic analysis of safety margin with normal action effects and log-normal resistance, in preparation.
Author’s
reply
I would like to express my appreciation of the Writers’ effort leading to further clarifying the FOTM Method, and thanks for helpful positive criticisms. The Writers’ comments on the limitations of the Taylor series application are certainly correct. It must be always kept in mind that the various applications of the HL-beta give only approximate information on the failure probabilities though the degree of agreement with the real failure probability is good. There are obviously many possibilities of developing the FOTM Method which might be of interest in the solution of advanced reliability problems. It is rather amazing that the FOTM Method had not been defined earlier, as it is a logical expansion of the original Hasofer’s and Lind’s idea. The Readers might be interested by the fact that the method is a result of “playing computer”. I only wanted to know what would happen if the non-normality of basic variables would be expressed by introducing the coefficients of skewness into the calculation model. I did not expect the phenomenon of the quasi-skewness invariance at all, and it took a lot of time to find the proof for this obvious feature. Erratum. Readers should kindly correct the following three-parameter log-normal probability distribution:
u,=&z+b,‘/” + la -f3p3+ a
1).
equation in the Appendix
on the
(A-4)