Method for identifying a most-probable-point in original space and its application for calculating failure probabilities

The present invention is a method for identifying a Most-Probable-Point (MPP) in original space obviating the need for a probability transformation, for use in first order/second order reliability analysis. It comprises generating the linear approximation of a limit-state-function, g(x) about the median, mean point, or mode of random variables, x of g(x). g(x) is defined so that g(x)f0 denotes a failure set. The Most-Probable-Point-Locus (MPPL) of g.sub.1, MPPL.sub.1, is constructed by the steps of: i) identifying the mode of x; ii) identifying the MPP of g.sub.1 (x)=c, where c is an arbitrary constant, and iii) constructing said MPPL.sub.1 by connecting said mode of x from step i), above, and the MPPs corresponding to different c's from step ii), above. A quadratic search algorithm is to identify point MPP.sub.1, the intersection of MPPL.sub.1 and g(x)=0, based on the following convergence criteria: ##EQU1## The process is then stopped, unless the convergence criteria are not satisfied. g.sub.2, the linear approximation of the limit-state-function, g(x) about MPP.sub.1 is generated. MPP.sub.21, the Most-Probable-Point of g.sub.2 (x)=0 is located. The approximate MPPL of g.sub.2, MPPL.sub.2, is constructed by connecting the mode and the MPP.sub.21. The quadratic search algorithm is used to identify point MPP.sub.2,, the intersection of MPPL.sub.2 and g(x)=0 based on the convergence criteria. g.sub.f (x) is updated, where j=2, 3, . . . , m, where m is the total number of steps required for convergence.

BACKGROUND OF THE INVENTION 
1. Field of the Invention 
This invention relates to calculating failure probabilities and more 
particularly to a methodology that involves identifying the 
most-probable-point in the original X-space with an iteration procedure. 
2. Description of the Related Art 
Several methods of approximation for calculating failure probabilities have 
been developed. These methods of approximation are described in the 
following references: 
Ang, A. H.-S., and W. H. Tang, Probability Concepts in Engineering Planning 
and Design, II--Decision, Risk, and Reliability, John Wiley & Sons, Inc., 
New York, N.Y. 1984. 
Madsen, H. O., S. Krenk, and N. C. Lind, Methods of Structural Safety, 
Prentice-Hall, Inc., Englewood Cliffs, N.J., 1986. 
Der Kiureghian, A., H.-Z. Lin, and S. J. Hwang, "Second-Order Reliability 
Applicatons," Journal of Engineering Mechanics, Vol. 113, No.8, ASCE, 
1987, pp. 1208-1255. 
Khalessi, M. R., Y.-T. Wu, and T. Y. Torng, "A new Most-Probable-Point 
Search Procedure for Efficient Structural Reliability Analysis," 
Proceedings of the 32nd Structures, Structural Dynamics, and Materials 
Conference, Part 2, AIAA/ASME/ASCE/AHS/ASC, April 1991, pp. 1295-1304 
Breitung, K., "Asymptotic Approximations for Multinormal Integrals," 
Journal of Engineering Mechanics, Vol. 110, No. 3, ASCE, 1984, pp. 
357-366. 
A common algorithm in these methods involves the transformation from the 
original X-space to a standard, uncorrelated normal U-space. This is 
conducted by the probability transformation that preserves cumulative 
density functions (CDFs) and PDFs at the linearization points in the 
iteration procedure. In reliability analysis, it is necessary to carry out 
this probability transformation only at a finite number of linearization 
points. The well-known first-and-second-order reliability methods 
(FORM/SORM) replace the limit-state function with a tangent plane and a 
quadratic surface, respectively, at a point u* on the limit-state surface, 
known as the U-space MPP (most-probable-point), which is at the minimum 
distance from the origin in the standard normal space. 
The advantage of FORM/SORM analyses is that the major contribution to the 
failure probability comes from the vicinity of u* because the probability 
density decays exponentially with the distance from u*. However, because 
the probability transformation used in the conventional FORM/SORM analyses 
preserves the CDF at the linearization points, the U-space MPP is 
inconsistent with the true MPP, x*, which has the highest probability 
density function (PDF) in the failure domain, when one of the 
distributions of the basic variables is asymmetric. That is, by inverse 
probability transformation, the mapping of the U-space MPP, u*, in the 
original space is not the true MPP, x*. This can be easily shown from the 
fact that the origin in the U-space, which has the highest PDF in the 
standard normal space, is consistent with the median point instead of the 
mode point in the original space for asymmetric distributions. 
Moreover, a nonlinearity from the probability transformation is imposed in 
the limit-state function in the standard normal space. This results in 
extra iterations to identify the MPP if probability transformation is 
employed in the analysis. Therefore, for accuracy and efficiency, it is 
desirable to develop a methodology to identify the MPP, x*, without using 
the probability transformation, and using x* to calculate the reliability. 
SUMMARY OF THE INVENTION 
The present invention involves a new iteration procedure to identify the 
most-probable-point (MPP) that has the highest probability density 
function in the failure domain without using probability transformation 
and involves a method to calculate the failure probability using the 
X-space MPP. The proposed iteration procedure constructs the 
most-probably-point-locus (MPPL) of a linearized limit-state function 
starting from the mode and uses this MPPL to search the next linearization 
point. The iteration is converged to the MPP, where the contour of equal 
probability density function is tangent to the limit-state function, g(x), 
which is defined so that g(x)&lt;0 denotes the failure set. Because 
probability transformation is not used in the iteration, the proposed 
method provides an efficient way to identify the X-space MPP. 
Once the X-space MPP has been identified, the conventional 
first-and-second-order reliability methods can be used to compute the 
failure probability by using the X-space MPP and an alternative 
probability transformation. The X-space MPP is transformed by using the 
alternative probability transformation into the standard normal space and 
then replace the limit-state function with a polynomial at the transferred 
U-space MPP. The alternative probability transformation preserves the 
probability density function (PDF) and its tangent at the linearization 
point so that the transferred U-space MPP is also the highest PDF point in 
the failure domain in the standard normal space. The present methods 
require no extra computation of the limit-state function. They may be 
performed using a standard computer. 
Other objects, advantages and novel features of the present invention will 
become apparent from the following detailed description of the invention 
when considered in conjunction with the accompanying drawings.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
The following nomenclature chart is provided to aid the reader: 
______________________________________ 
P.sub.f probability of failure 
x vector of random variables 
n total number of random variables 
f.sub.x (x) 
joint probability density function (PDF) 
g (x) limit-state function 
vg (x) vector of the gradient of g (x) 
vf.sub.x (x) 
vector of the gradient of f.sub.x (x) 
g.sub.j (x) 
linearlized limit-state function obtained by 
linearizing g (x) at x.sub.j 
MPP most-probable-point (design point) 
MPPL most-probable-point-locus 
.mu. mean value 
x* vector of MPP 
.beta. distance to the mode in scaled X-space 
.epsilon..sub.g 
convergence tolerance in g 
.epsilon..sub..beta. 
convergence tolerance in .beta. 
x.sub.j1 MPP of g.sub.j (x) = 0 
MPPL.sub.j 
MPPL of g.sub.j (x) = 0 
.beta..sub.j1 
distance between x.sub.j1 and the mode 
.beta..sub.j 
distance from the intersection of MPPL.sub.j and 
g (x) = 0 to the mode 
T.sub.0 a feasible search direction 
x.sub.0 a starting point to find x.sub.j1 
S Search direction to find x.sub.j1 
.alpha..sub.0 
vg (xg)/.vertline.v.sub.g (x.sub.0).vertline. 
.PHI.(u.sub.i) 
CDF of standard normal variable u.sub.i 
.phi. PDF of standard normal distribution 
x.sub.i-1 partial vector, = {x.sub.1, x.sub.2, . . . x.sub.i-1 }.sup.T 
.sigma. standard deviation 
.mu.' equivalent mean value 
.sigma.' equivalent standard deviation 
______________________________________ 
1. Method for Identifying the X-Space MPP, x* 
Determination of the X-space MPP, x*, of a limit-state function without 
using the probability transformation is an optimization problem with a 
nonlinear constraint. This problem can be represented by 
##EQU2## 
where R.sup.n denotes the domain of the X-space. 
Equation 1 indicates that the contour of equal probability is tangent to 
the limit-state surface at the MPP,x*, because x* has the highest 
probability density in the failure domain (FIG. 1). If all random 
variables are standard normal distributed, the gradient of PDF passes 
through the origin because the contours of equal probability are 
concentric hyperspheres. This conclusion coincides with the criterion used 
in the iteration procedure to identify the U-space MPP,u*. 
A problem in the X-space approach is that the number of iterations will be 
affected by the units of basic random variables. An easy way to solve this 
problem is to scale down the dimensions by the corresponding standard 
deviation. This procedure does not introduce more nonlinearity in the 
limit-state function because it is a linear transformation. The scaled 
X-space, then, becomes a "standard" space with the same scale for all 
dimensions and the result is invariant to the units of random variables. 
Because the evaluation of the limit-state function is expensive such as in 
finite-element-based reliability analysis used in the structural design, 
the iteration procedure for the identification of the MPP should require a 
minimal number of limit-state function calculations. A fast convergence 
algorithm always involves the gradient of the limit-state function (a type 
of steepest descent method). Calculating the gradient of the limit-state 
function is the major computational effort in many structural systems. 
Because of this, we can use the linearized limit-state function generated 
at the current linearization point x.sub.j to approach the solution x*. 
The advantage of this approach is that computational effort involved in 
the linearized limit-state function is trivial, since no finite-element 
analysis is required. The linearized limit-state function at the 
linearization point x.sub.j is obtained from the following equation: 
EQU g.sub.j (x)=g(x.sub.j)+.gradient.g.sup.T (x.sub.j)(x-x.sub.j)(2) 
By intuition, the MPP of the linearized limit-state function generated at 
x.sub.j, denoted as g.sub.j (x)=g(x.sub.j), is closer to x* than x.sub.j. 
This is proved later. Finding this MPP is also an optimization problem 
with a linear constraint. By replacing g(x)=0 with g.sub.j (x)=g(x.sub.j), 
Equation 1 can be used to model this problem. The variable metric method 
described in Philipe, E. G., M. Walter, and H. W. Margaret, Practical 
Optimization, Academic Press, Inc., San Diego, Calif., 1981, can be 
employed to find the MPP of the linearized limit-state function g.sub.j 
(x)=g(x.sub.j). Repeating the same procedure, the MPP of the linearized 
limit-state function g.sub.j (x)=c can be identified, where c is a 
specified constant. The MPPL of a linearized limit-state function g.sub.j 
(x), MPPL.sub.j, is then obtained by connecting the MPPs of the linearized 
limit-state surfaces g.sub.j (x) corresponding to different selected c. 
Note that the construction of MPPL.sub.j does not need the extra 
limit-state function as well as its gradient calculations. The 
computational effort required to construct this MPPL is then insignificant 
because the required PDF and its gradient calculations (as shown below) 
are minor compared to limit-state function calculation. 
At the MPP of the linearized limit-state function g.sub.j (x)=c, the 
contour of equal PDF is tangent to g.sub.j (x)=c. The gradient of g.sub.j 
(x) is a constant and the contours of equal probability are not concentric 
hyperspheres in the original space if one of the basic random variables is 
asymmetric. Therefore, the MPPL.sub.j is a curve if one of basic random 
variables is asymmetric. Also, for a given linearized limit-state function 
g.sub.j (x), the MPPL constructed in the original space is different from 
the MPPL constructed in the standard normal space because of the 
inconsistency of the MPPs in the two spaces if the conventional 
probability transformation is used. Furthermore, because the mode is an 
MPP for any linearized limit-state function, the MPPL can constructed from 
the mode. 
The quadratic search algorithm (see, for example, the Khalessi et al. 
reference previously cited) can be then used to find the g(x)=0 point 
along MPPL.sub.j ; that is, the quadratic search algorithm is applied to 
find the intersection of g(x)=0 and the MPPL.sub.j. Once this intersection 
has been identified, a new linearized limit-state function is created by 
linearizing the limit-state function at this intersection point. This 
updated linearized limit-state function is then used to construct the 
updated MPPL. This procedure is repeated until the convergency of x* is 
reached. 
2. Iteration Procedure for Identifying X-Space MPP, x* 
As indicated in the previous section, the MPPL.sub.j of the linearized 
limit-state function, g.sub.j (x), is constructed through the following 
procedure: 
1. Identify the mode that has the highest PDF in the original space. 
2. Linearize g(x) at current linearization point x.sub.j, i.e., g.sub.j 
(x)=g (x.sub.j)+.gradient.g.sup.T (x.sub.j) (x-x.sub.j). 
3. Identify the MPP of g.sub.j (x)=c, where c is an arbitrary constant, by 
an optimization algorithm such as the variable metric method. (See, for 
example, the Philipe et al. reference previously cited.) 
4. Obtain the MPPL.sub.j for g.sub.j (x) by connecting the mode and the 
MPPs from Step 3. 
The following is the proposed iteration procedures to identify the X-space 
MPP,x* (see FIG. 2): 
1. Divide each random variable by its standard deviation. 
2. Construct the first MPPL, MPPL.sub.1, for the linearized g(x) at mean 
point x.sub.u through the above procedures. 
3. Use the quadratic search algorithm to identify the intersection of 
g(x)=0 and MPPL.sub.1. Denote this intersection point as x.sub.2. 
4. Establish the linearized g(x) at x.sub.2, i.e., g.sub.2 
(x)=g(x.sub.2)+.gradient.g.sup.T (x.sub.2) (x-x.sub.2). 
5. Check convergence criteria (see Section 3). If convergent, stop; 
otherwise, continue to Step 6. 
6. Approximate MPPL.sub.2 as a line that passes through the mode and the 
MPP of g.sub.2 (x)=0. 
7. Use the quadratic search algorithm to identify the intersection of 
g(x)=0 and MPPL.sub.2. Denote this intersection point as x.sub.3. 
8. Repeat steps 4 through 7 until convergence criteria are met. 
Although MPPL of the linearized limit-state-function g.sub.2 (x) is also a 
curve, it is approximated by a straight line passing through the mode and 
the MPP of g.sub.2 (x)=0. The effect of this approximation on the search 
for the intersection of MPPL.sub.2 and g(x)=0 is insignificant because the 
MPP of g.sub.2 (x)=0 is close to the limit-state surface. Similarly, for 
any linearized limit-state function with the linearization point close to 
the limit-state surface, its MPPL can be assumed to be a straight line 
passing through the mode and the corresponding MPP of the linearized 
limit-state function. 
3. Convergence Criteria for the Iteration Procedure to Identify X-Space MPP 
As mentioned in Section 1, the contour of equal probability density 
function is tangent to the limit-state surface at the MPP,x*. In other 
words, x* satisfies the following criteria: 
##EQU3## 
where .gradient.f.sub.x (x*) and .gradient.g(x*) are the first-order 
derivatives of the PDF and the limit-state function, respectively, with 
respect to the basic random variables x evaluated at x*. 
Because the present iteration procedure for the identification of the MPP 
is based in part on the concept of the MPPL reliability method (see 
Khalessi, M. R., Y.-T. Wu, and T. Y. Torng, "A New Most-Probable-Point 
Search Procedure for Efficient Structural Reliability Analysis," 
Proceedings of the 32nd Structures, Structural Dynamics, and Materials 
Conference, Part 2, AIAA/ASME/ASCE/AHS/ASC, April 1991, pp. 1295-1304), 
the following two loose convergent criteria can be also used: 
EQU .vertline.g(x.sub.j1).vertline..ltoreq..epsilon..sub.g (5) 
where x.sub.j1 is the MPP of linearized limit-state function g.sub.j (x)=0 
and .epsilon..sub.g is a tolerance in checking g-function; and 
EQU .vertline..beta..sub.j1 -.beta..sub.j 
.vertline..ltoreq..epsilon..sub..beta.(6) 
where .beta..sub.j1 is the distance between x.sub.j1 (the MPP of g.sub.j 
(x)=0) and the mode, .beta..sub.j, is the distance from the intersection 
point of MPPL.sub.j and the limit-state function to the mode, and 
.epsilon..sub..beta. is the tolerance in checking .beta.. 
4. Proof of Convergence of Proposed Method 
For an optimization problem, any point that satisfies all the constraints 
is said to be feasible. The set of all feasible points is called the 
feasible domain. In structural reliability analysis, the feasible domain 
is equivalent to the failure domain. A feasible search direction T.sub.o 
based on the following two definition is chosen: 
1. A vector T.sub.o is a feasible direction from the point x.sub.o if at 
least a small step can be taken along it that does not immediately leave 
the feasible domain. 
2. A vector T.sub.o is a useful feasible direction from the point x.sub.o 
if, in addition to Definition 1, T.sub.o.sup.T .gradient.f.sub.x 
(x.sub.o)&gt;0. 
The first definition indicates that T.sub.o.sup.T 
.gradient.g(x.sub.o).ltoreq.0 for a concave limit-state function. These 
two definitions ensure that point x.sub.o moves forward to the solution x* 
along the feasible direction T.sub.o. 
The proof of the convergence of the subject algorithm is illustrated in 
FIG. 3. The direction starting from the linearization point x.sub.j to the 
MPP of the linearized limit-state function g.sub.j (x)=g(x.sub.j) (=0, in 
this figure) is defined as T.sub.j. In the case of a concave limit-state 
function, T.sub.j is a feasible direction because it satisfies the 
preceding two definitions. Furthermore, because the linearized limit-state 
surface g.sub.j (x)=g(x.sub.j) is in the failure domain, its corresponding 
MPP is between the linearized point x.sub.j and x*. Therefore, the 
intersection point x.sub.j+1 of MPPL.sub.j and the limit-state surface 
g(x)=0 is closer to x* than the linearized point x.sub.j. In the case of a 
convex limit-state surface, T.sub.j does not satisfy the first condition 
of feasible direction and x* is located between the linearized point 
x.sub.j and the MPP of the linearized limit-state surface g.sub.j 
(x)=g(x.sub.j). However, the intersection point x.sub.j+1, of the 
limit-state surface and MPPL.sub.j has a higher PDF than the linearized 
point x.sub.j, since at 
x.sub.j+1,.gradient.g(x.sub.j)//.gradient.fx(x.sub.j+1) and 
g(x.sub.j+1)=0. Thus, x.sub.j+1 is closer to x* than the linearized point, 
x.sub.j. Therefore, from preceding discussion, we conclude that the 
proposed iteration algorithm will converge to the MPP, x*. 
5. Alternative Probability Transformation 
Once the X-space MPP, x* is identified, the reliablity can be calculated by 
using this point. For this purpose, the following alternative probability 
transformation is derived. According to Horne, M. R., and P. H. Price, 
"Commentary on the Level II Procedure," in Rationalization of Safety and 
Serviceability Factors in Structural Codes, Report No. 63, Construction 
Industry Research and Information Association, London, 1977, pp. 209-271, 
if the probability transformation preserves the PDF and first-order 
differentials of the PDF at the linearization point, the U-space MPP, u*, 
will be consistent with the X-space MPP, x*. For independent random 
variables, this alternative probability transformation is derived as 
follows: Because the alternative probability transformation preserves the 
PDF at x*, 
EQU f.sub.u.sbsb.i (u.sub.i.sup.*)du.sub.i +f.sub.x.sbsb.i 
(x.sub.I.sup.*)dx.sub.i (7) 
where f.sub.u.sbsb.i (u.sub.i) denotes the PDF of a standard normal 
distribution. As shown in FIG. 4, at the X-space MPP, we can replace the 
distribution of variable x.sub.i with a normal distribution which has an 
equivalent mean and standard deviation, denoted as 
EQU .mu..sub.i.sup.'* and .sigma..sub.i.sup.'*, respectively. 
That is, we have the following probability transformation: 
##EQU4## 
Then, Equation 7 becomes 
##EQU5## 
From the preservation of the first-order derivative of the PDF and 
equations 8 and 9, we have the following: 
##EQU6## 
Eliminating .sigma..sub.i.sup.'* between equations 9 and 10 leaves 
##EQU7## 
where f.sub.x.sbsb.i.sup.' (x.sub.i.sup.*)) is the first-order derivative 
of f.sub.x.sbsb.i evaluated at the X-space MPP, x.sub.i *. 
The preceding equation gives an alternative probability transformation for 
the cases of independent basic random variables. For a given x.sub.i * the 
corresponding u.sub.i * can be computed by the preceding equation. After 
obtaining u.sub.i *, the equivalent mean, .mu..sub.i.sup.'*, and standard 
deviation, .sigma..sub.i.sup.'*, can be computed from equations 8 and 9, 
respectively. 
By definition, the X-space MPP, x*, and the transferred U-space MPP, u*, 
have the highest PDF in the failure domain of the original and standard 
normal space, respectively. Therefore, to prove the consistency of x* and 
u* through the preceding probability transformation, it is necessary to 
show that the gradient of the PDF is parallel to the gradient of the 
limit-state function at both x* and u*. That is, 
##EQU8## 
From Equation 9, 
##EQU9## 
Equations 9 through 14, after some vector operations, yield 
##EQU10## 
where 
##EQU11## 
is the diagonal Jacobian matrix for the independent basic random 
variables. By the chain rule, 
##EQU12## 
Equations 15 and 16 show that if .gradient.f.sub.x (x*)//.gradient.g(x*), 
then .gradient.f.sub.u (u*)//.gradient.g(u*). In other words, this 
alternative probability transformation yields consistency between x* and 
u*. 
Note that the preceding alternative probability transformation produces a 
limit-state function in the standard normal space that is different from 
the one used in the conventional FORM/SORM analyses, which is denoted as 
u.sub.R *. Therefore, although both u* and u.sub.R * have the highest PDF 
in the failure domain in the standard normal space, they are different 
points. 
Similarly, for the case of dependent basic variables with known joint PDF, 
by using the Rosenblatt's transformation described in Rosenblatt, M., 
"Remarks on a Multivariate Transformation," Annals Math. Stat., Vol. 23, 
1952, pp. 470-472, the following probability transformation can be 
derived: 
##EQU13## 
As with the conventional Rosenblatt's transformation, the transferred 
U-space MPP, u*, depends on the ordering of the basic random variables, x. 
However, because both x* and u* have the highest PDF and they are 
consistent through the alternative probability transformation, u* can be 
used in the important sampling technique for evaluating failure 
probability. 
6. Calculation of Failure Probability 
Once the transferred U-space MPP, u*, has been located, the strategy used 
in the conventional FORM/SORM methods can be used to approximate the 
failure probability which is defined as follows: 
##EQU14## 
That is, the limit-state surface at the U-space MPP, u*, is replaced with a 
tangent plane for the FORM method and a quadratic surface for the SORM 
method. Then failure probability for the FORM method is 
EQU p.sub.f .apprxeq.p.sub.f.sbsb.1 =.phi.(-.beta.) (21) 
where .beta. is the distance of the transferred U-space MPP, u*, from the 
origin in the standard normal space. 
The formula for failure probability in the SORM method is given by Tvedt, 
L., "Distribution of Quadratic Forms in Normal Space--Application to 
Structural Reliability," Journal of Engineering Mechanics, Vol. 116, No. 
6, ASCE, June 1990, pp. 1183-1192. It calculates the failure probability 
by replacing the limit-state surface with a quadratic surface in the 
standard normal space. That is, p.sub.f .apprxeq.p.sub.f.sbsb.2, where 
p.sub.f.sbsb.2 equals 
##EQU15## 
and .phi. is the PDF of a normal distribution, Re is the real part of a 
complex number, i={-1}.sup.1/2 and k.sub.j are the main curvatures of the 
limit-state function at the U-space MPP in the rotated standard normal 
space. The integration path of the preceding equation is shifted parallel 
using the saddle point method into one passing through a saddle point, and 
the trapezoidal rule is applied to compute the integral. 
The curvature matrix of the limit-state function in the rotated standard 
normal space, C, is computed by 
##EQU16## 
where R is an orthogonal matrix used to rotate the standard normal U-space 
to a rotated standard U'-space and D is the second derivative of the 
limit-state function at u*. R is selected such that the nth row of R is 
the unit normal of the limit-state function at the U-space MPP, i.e., 
-.gradient.g(u*)/.vertline..gradient.g(u*).vertline.. The Gram-Schmidt 
algorithm can be used to generate R. 
The major effort in the SORMmethod is computing the curvature matrix of the 
limit-state function at the transferred U-space MPP. Usually, the 
finite-difference scheme is directly applied in the rotated standard 
normal space to compute the curvature matrix, Der Kiureghian, A., H.-Z. 
Lin, and S.-J. Hwang, "Second-Order Reliability Approximations," Journal 
of Engineering Mechanics, Vol. 113, No. 8, ASCE, 198, pp. 1208-1225. The 
curvature matrix C in Equation 23 is an n-1.times.n-1 matrix. A 
central-difference scheme requires a total of 2(n-1).sup.2 computations of 
the limit-state function (this number is reduced to n(n+1)/2, if a 
forward-difference scheme is used). It is obvious the computation of this 
curvature matrix must be simplified to perform the SORM analysis in 
finite-element-based reliability analysis. 
In some cases, it is advantageous to compute the second-derivative matrix 
in the original space and then make the transformation to compute the 
curvature matrix in the standard normal space. For example, if the 
limit-state function in the original space is linear, curvatures at the 
U-space MPP arise only because of nonlinearity in the probability 
transformation. Therefore, the following improvement is made to reduce 
computational effort: 
##EQU17## 
From the preceding equation, curvatures of the limit-state function in the 
U-space are contributed from two parts: the curvatures of the limit-state 
function in the X-space and the nonlinearility of the probability 
transformation. Equation 24 requires a full curvature matrix of the 
limit-state function, that is, an n.times.n matrix in both X- and 
U-spaces. Therefore, it is used only in the case where the computation of 
the curvature matrix requires less finite-element analysis in the X-space 
than in the U-space. If the second-order derivative of the limit-state 
function in the X-space in the preceding equation is eliminated, then 
##EQU18## 
The preceding equation shows that only curvatures of the limit-state 
function in the standard normal space contributed by the nonlinearity of 
the probability transformation are counted. In other words, the 
limit-state surface is replaced by a hyper-tangent plane at the X-space 
MPP, x*. The second order derivative of the probability transformation can 
be computed by a finite-difference scheme or by tensor analysis. Tensor 
analysis to compute the curvature matrix is useful in the case of 
dependent random variables. 
The gradient of the limit-state function at the X-space MPP in the 
preceding equation is available in the iteration procedure to identify the 
X-space MPP. Therefore, with minor extra work and without extra 
finite-element analysis, we can obtain an improvement in failure 
probability over the first-order failure probability (Equation 21). 
EXAMPLE 
This is an example of local buckling stress for a flanged component 
(Structural Manual, Vol. 3, Space Systems Division, Rockwell 
International, September 1974) under longitudinal compression. The 
limit-state function is 
EQU g(x)=.phi..sub.cr -.phi. (26) 
where .phi..sub.cr is the local buckling stress for flanges under here 
ongitudinal compression and .phi. is the stress response computed from 
finite-element analysis. .phi..sub.cr can be computed by 
##EQU19## 
where .eta. is the plasticity correction, .mu. is Poisson's ratio, E is 
Young's modulus, K.sub.L is the local buckling stress coefficient, and 
t.sub.L and b.sub.L are defined in FIG. 5. K.sub.L can be obtained from 
the chart of K.sub.L versus b.sub.L /b.sub.f shown in FIG. 5, which was 
determined from test results. K.sub.L can be represented as a regression 
equation: 
##EQU20## 
where c.sub.1 and c.sub.2 are regression coefficients and estimated as 
-0.2421 and 2.1053, respectively, from FIG. 6. To demonstrate the proposed 
method, the assumed basic variables x are given in Table 1. 
TABLE 1 
______________________________________ 
Statistical Properties of the Basic 
Random Variables for Example 
Random Distribu- Standard 
Variable 
Name tion Type Mean Deviation 
______________________________________ 
x.sub.1 
h Weibull 2.5 0.75 
x.sub.2 
E Gamma 3 .times. 10.sup.7 
6 .times. 10.sup.6 
x.sub.3 
m Gamma 2.5 .times. 10.sup.-1 
7.5 .times. 10.sup.-2 
x.sub.4 
t.sub.L Lognormal 5 .times. 10.sup.-2 
5 .times. 10.sup.-4 
x.sub.5 
b.sub.L Lognormal 4.75 .times. 10.sup.-1 
4.75 .times. 10.sup.-2 
x.sub.6 
b.sub.f Lognormal 1.45 1.45 .times. 10.sup.-2 
x.sub.7 
s Gamma 1 .times. 10.sup.5 
2.5 .times. 10.sup.4 
x.sub.8 
c.sub.1 Constant -2.42113 .times. 10.sup.-1 
0 
x.sub.9 
c.sub.2 Constant 2.1053 0 
______________________________________ 
Convergence criteria from equations 3 and 4 are used in this example. 
Furthermore, we assume the tolerance for these two criteria as 
1.times.10.sup.-5 and the maximum steps in the quadratic search algorithm 
for searching g(x)=0 point along MPPL.sub.j as 2. The result is listed in 
Table 2. 
TABLE 2 
______________________________________ 
Results for Example 
Number Number 
of of 
Iteration g (x) yg (x) 
______________________________________ 
1 4 1 
2 3 1 
3 0 1 
______________________________________ 
Seven limit-state functions and three gradient computations are required to 
obtain the MPP, x*, with the coordinates (1.039, 2.516.times.10.sup.7, 
2.213.times.10.sup.-1, 4.996.times.10.sup.-2, 4.740.times.10.sup.-1, 
1.450, 1.120.times.10.sup.5, -2.421.times.10.sup.-1, 2.105). The PDF at x* 
is 6.646.times.10.sup.-8. 
For comparison, this example was rerun using the HL-RF method and a 
tolerance value 0.001. Eight limit-state functions and eight gradient 
computations were used to obtain the conventional U-space MPP, u*. The 
corresponding inverse coordinates in the X-space were (1.052, 
2.594.times.10.sup.7, 2.346.times.10.sup.-1, 4.997.times.10.sup.-2, 
4.794.times.10.sup.-1, 1.451, 1.170.times.10.sup.5, 
-2.421.times.10.sup.-1, 2.105). The PDF at this point is 
6.131.times.10.sup.-8. This shows that the U-space MPP is not the true 
MPP, and the proposed iteration procedure provides an efficient way to 
find the X-space MPP without using probability transformation. 
The resulting X-space MPP, x*, is then used to compute failure probability 
by the first-order reliability method (Equation 21), and the simplified 
second-order reliability method that neglects curvatures of the 
limit-state function in the X-space (equations 22 and 25). These two 
methods use the alternative probability transformation (equations 11 and 
17) because the X-space MPP is used to compute failure probability. For 
ease of identification, the last two methods are called FORM/A and 
Simplified SORM/A. Because FORM/A and Simplified SORM/A do not require 
computing curvatures of the limit-state function in the X-space, these two 
methods are very efficient. 
Failure probability results for this example using the X-space MPP are 
listed in Table 3. 
TABLE 3 
______________________________________ 
Results of Example 
Method P.sub.f 
______________________________________ 
FORM/A 1.705 .times. 10.sup.-2 
Simplified SORM/A 1.593 .times. 10.sup.-2 
______________________________________ 
These results are compared with failure probability computed through Monte 
Carlo simulation, which is 1.616.times.10.sup.-2, and the corresponding 
coefficient of variation is 4.93.times.10.sup.-2 with 25,000 trials. The 
result of the Monte Carlo simulation can be considered the exact value; 
therefore, this example shows the efficiency and accuracy of the invented 
method. 
Referring now to FIG. 7, a flow chart of the method for identifying a 
Most-Probable-Point (MPP) in original space in accordance with the 
principles of the present invention is illustrated. 
Obviously, many modifications and variations of the present invention are 
possible in light of the above teachings. It is, therefore, to be 
understood that, within the scope of the appended claims, the invention 
may be practiced otherwise than as specifically described.