- Email: [email protected]
On the choice of expansion point in form or sorm
Structural Safety, 4 (1987) 243-245 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
243
SHORT COMMUNICATION
ON THE CHOICE OF EXPANSION POINT IN FORM OR SORM Ove DiUevsen Department of Structural Engineering, Technical University of Denmark, DK-2800 Lyngby (Denmark)
(Received December 25, 1985; accepted June 10, 1886)
The problem of choosing the best single point on the limit state surface for first or second order expansion of the surface in order to make F O R M or SORM approximations to the failure probability is still under discussion. The paper by Pearce and Wen [1] adds some illustrative examples for the case of the time varying problem where the concern is the calculation of the mean outcrossing rate ~,(D) from the safe set D. Three natural expansion point candidates are considered: (1) the point closest to the origin; (2) the point of maximum local mean outcrossing rate; (3) a stationarity point of the mean outcrossing rate out of the tangent hyperplane as the tangent point is varying over the limit state surface. These characterizations of the points all refer to a representation of the problem in a standardized Gaussian space corresponding to fixed time. In all examples the third choice is the best but unfortunately the authors' general argumentation for the thesis that the third choice is superior to all other points at least for convex safe sets or for convex failure sets is not valid. In fact, it is not true that the mean outcrossing rate out of a subset of the safe set is guaranteed to be an upper bound on the mean outcrossing rate out of the safe set. However, the argument applies correctly to the left side of the inequality
Pr(T) <~Pr(0) + v( D)V
(1)
For a convex failure set (but not necessarily for a convex Safe set) the right side of the inequality applied to any tangent hyperplane therefore gives an upper bound. Thus the best choice of tangent point for F O R M calculation of the right side of the inequality corresponds to the tangent plane for which the F O R M result is minimized. For any hyperplane in distance fl from the origin and for a stationary Gaussian vector process X with differentiable sample paths the right side of the inequality is T (I)(-fl)+ 2-~-vep(/~)~
(21
provided X(t) for fixed t has unit covariance matrix. In this formula • and cp are the standardized Gaussian distribution function and density function, respectively, C is the covari0167-4730/87/$03.50
© 1987 Elsevier Science Publishers B.V.
244 ance matrix of the derivative process X'(t) for fixed t, while a is the unit normal vector to the hyperplane. Assume that the hyperplane is tangential to the limit state surface at a point Q and let s be an arch length parameter on a curve through Q on the surface such that s = 0 at Q. If Q is a stationarity point for (2) the derivative of (2) with respect to s must be zero for s = 0. This gives the equation
( 2¢~ + T13v/a-TC--d)-~-~s = T d s
(3)
It follows from this that if the right side is non-zero, then d13/ds is non-zero. This excludes the point closest to the origin as being the best approximation point. For large values of T the term ~/2¢r becomes small as compared to the second term in the parenthesis. This leads to the choice of approximation point suggested by Pearce and Wen [1]. It is interesting to study the asymptotic behavior of (3) for k---, m when the safe set is multiplicatively expanded by a factor k about the origin. Then (2) is replaced by T ¢ ( - k13) + 2-¢-Gj~ (k13)CUd-g
(4)
which by differentiation with respect to s gives the equation
(s) Since the right side of this equation is independent of k it follows that d B / d s ~ 0 for k ~ m. Thus the point closest to the origin is asymptotically the best linearization point as k ---, ~ . Of course, this is consistent with recent interesting results by Breitung [2] and Hohenbichler [3]. They show that the mean outcrossing rate out of suitably regular safe sets is determined asymptotically in explicit form for k ~ ~ solely by the distance fl of the closest point to the origin and the local first and second order geometrical properties of the limit state surface at this point. The results are valid in case the point is a regular point with respect to differentiability of the limit state surface or a simple type singular point in the sense that the singularity is defined by a finite n u m b e r of linearly independent normal vectors to the surface at the point. Besides the value of 13 it is the set of normal vectors and the curvature measures in the directions orthogonal to the set of normal vectors that determine the mean outcrossing rate asymptotically as k --* m. Similar asymptotic results are valid for the instantaneous probability of failure Pf (0). Their role in particular in system reliability analysis by use of F O R M or S O R M is discussed in an important forthcoming paper by Hohenbichler et al. [4]. It is noted from the limit passage of (5) for k ~ oe that there are several expansion points that lead to the asymptotically correct result as k ~ m. Both the points 1, 3, and the point obtained as solution to (3) have this property. At least for the case with convex failure set the last point gives the best F O R M approximation to the failure probability pf (T). For other cases than this there seems to be no clear criterion for the choice of a uniquely characterized expansion point except, perhaps, that it is heuristically reasonable to choose the point among those that are consistent with the asymptotic analysis. At present only experience can guide the choice. For the large values of 13 relevant in structural reliability the problem is usually less critical. Therefore, a simplicity principle favours the point closest to the origin.
245
REFERENCES 1 H.T. Pearce and Y.-K. Wen, On linearization points for nonlinear combination of stochastic load processes. Structural Safety, 2 (1985) 169-176. 2 K. Breitung, Asymptotic approximations for multinormal domain and surface integrals. Fourth International Conference on Applications of Statistics and Probability in Soil and Structural Engineering. Universit~ di Firenze (Italy), 1983, Pitagora Editrice, Vol. 2, pp. 755-767. 3 M. Hohenbichler, An asymptotic formula for the crossing rate of normal processes into intersections. Berichte zur Zuverl~issigkeitstheorie der Bauwerke, SFB 96 der TU Miinchen, Heft 75, Miinchen 1984. 4 M. Hohenbichler, S. Gollwitzer, W. Kruse and R. Rackwitz, New light on first- and second-order reliability methods. Manuscript submitted to Structural safety, TU Miinchen, 1985.
243
SHORT COMMUNICATION
ON THE CHOICE OF EXPANSION POINT IN FORM OR SORM Ove DiUevsen Department of Structural Engineering, Technical University of Denmark, DK-2800 Lyngby (Denmark)
(Received December 25, 1985; accepted June 10, 1886)
The problem of choosing the best single point on the limit state surface for first or second order expansion of the surface in order to make F O R M or SORM approximations to the failure probability is still under discussion. The paper by Pearce and Wen [1] adds some illustrative examples for the case of the time varying problem where the concern is the calculation of the mean outcrossing rate ~,(D) from the safe set D. Three natural expansion point candidates are considered: (1) the point closest to the origin; (2) the point of maximum local mean outcrossing rate; (3) a stationarity point of the mean outcrossing rate out of the tangent hyperplane as the tangent point is varying over the limit state surface. These characterizations of the points all refer to a representation of the problem in a standardized Gaussian space corresponding to fixed time. In all examples the third choice is the best but unfortunately the authors' general argumentation for the thesis that the third choice is superior to all other points at least for convex safe sets or for convex failure sets is not valid. In fact, it is not true that the mean outcrossing rate out of a subset of the safe set is guaranteed to be an upper bound on the mean outcrossing rate out of the safe set. However, the argument applies correctly to the left side of the inequality
Pr(T) <~Pr(0) + v( D)V
(1)
For a convex failure set (but not necessarily for a convex Safe set) the right side of the inequality applied to any tangent hyperplane therefore gives an upper bound. Thus the best choice of tangent point for F O R M calculation of the right side of the inequality corresponds to the tangent plane for which the F O R M result is minimized. For any hyperplane in distance fl from the origin and for a stationary Gaussian vector process X with differentiable sample paths the right side of the inequality is T (I)(-fl)+ 2-~-vep(/~)~
(21
provided X(t) for fixed t has unit covariance matrix. In this formula • and cp are the standardized Gaussian distribution function and density function, respectively, C is the covari0167-4730/87/$03.50
© 1987 Elsevier Science Publishers B.V.
244 ance matrix of the derivative process X'(t) for fixed t, while a is the unit normal vector to the hyperplane. Assume that the hyperplane is tangential to the limit state surface at a point Q and let s be an arch length parameter on a curve through Q on the surface such that s = 0 at Q. If Q is a stationarity point for (2) the derivative of (2) with respect to s must be zero for s = 0. This gives the equation
( 2¢~ + T13v/a-TC--d)-~-~s = T d s
(3)
It follows from this that if the right side is non-zero, then d13/ds is non-zero. This excludes the point closest to the origin as being the best approximation point. For large values of T the term ~/2¢r becomes small as compared to the second term in the parenthesis. This leads to the choice of approximation point suggested by Pearce and Wen [1]. It is interesting to study the asymptotic behavior of (3) for k---, m when the safe set is multiplicatively expanded by a factor k about the origin. Then (2) is replaced by T ¢ ( - k13) + 2-¢-Gj~ (k13)CUd-g
(4)
which by differentiation with respect to s gives the equation
(s) Since the right side of this equation is independent of k it follows that d B / d s ~ 0 for k ~ m. Thus the point closest to the origin is asymptotically the best linearization point as k ---, ~ . Of course, this is consistent with recent interesting results by Breitung [2] and Hohenbichler [3]. They show that the mean outcrossing rate out of suitably regular safe sets is determined asymptotically in explicit form for k ~ ~ solely by the distance fl of the closest point to the origin and the local first and second order geometrical properties of the limit state surface at this point. The results are valid in case the point is a regular point with respect to differentiability of the limit state surface or a simple type singular point in the sense that the singularity is defined by a finite n u m b e r of linearly independent normal vectors to the surface at the point. Besides the value of 13 it is the set of normal vectors and the curvature measures in the directions orthogonal to the set of normal vectors that determine the mean outcrossing rate asymptotically as k --* m. Similar asymptotic results are valid for the instantaneous probability of failure Pf (0). Their role in particular in system reliability analysis by use of F O R M or S O R M is discussed in an important forthcoming paper by Hohenbichler et al. [4]. It is noted from the limit passage of (5) for k ~ oe that there are several expansion points that lead to the asymptotically correct result as k ~ m. Both the points 1, 3, and the point obtained as solution to (3) have this property. At least for the case with convex failure set the last point gives the best F O R M approximation to the failure probability pf (T). For other cases than this there seems to be no clear criterion for the choice of a uniquely characterized expansion point except, perhaps, that it is heuristically reasonable to choose the point among those that are consistent with the asymptotic analysis. At present only experience can guide the choice. For the large values of 13 relevant in structural reliability the problem is usually less critical. Therefore, a simplicity principle favours the point closest to the origin.
245
REFERENCES 1 H.T. Pearce and Y.-K. Wen, On linearization points for nonlinear combination of stochastic load processes. Structural Safety, 2 (1985) 169-176. 2 K. Breitung, Asymptotic approximations for multinormal domain and surface integrals. Fourth International Conference on Applications of Statistics and Probability in Soil and Structural Engineering. Universit~ di Firenze (Italy), 1983, Pitagora Editrice, Vol. 2, pp. 755-767. 3 M. Hohenbichler, An asymptotic formula for the crossing rate of normal processes into intersections. Berichte zur Zuverl~issigkeitstheorie der Bauwerke, SFB 96 der TU Miinchen, Heft 75, Miinchen 1984. 4 M. Hohenbichler, S. Gollwitzer, W. Kruse and R. Rackwitz, New light on first- and second-order reliability methods. Manuscript submitted to Structural safety, TU Miinchen, 1985.