A DYNAMIC INVERSE FORM METHOD: DESIGN ...

A DYNAMIC INVERSE FORM METHOD: DESIGN CONTOURS FOR LOAD COMBINATION PROBLEMS Loren D. Lutesa,∗*, Steven R. Wintersteinb a

Zachary Department of Civil Engineering, Texas A&M University, 105 Greenway Dr., Brenham TX 77833, USA. b Probability-Based Engineering, 610 Gilbert Ave., Menlo Park, CA 94025, USA. In static structural reliability problems involving a set of random variables, it is common to relate the failure probability to the event that a safety margin becomes negative. For dynamic responses we usually seek the mean out-crossing rate into the unsafe region. Standard FORM (First-Order Reliability) methods can be applied to both the static and dynamic problems. A challenge comes in the design context, when a target probability of failure or crossing rate is specified and a particular design variable (e.g., capacity) is sought. In the static case, inverse FORM methods have been suggested. The resulting "design contour" may be viewed, in the transformed space of standard Gaussian variables, as the locus of all possible FORM design points that can give the desired probability of failure value. These contours have been previously used for the static problem to determine "design contours" for various problems. We here develop analogous design contours for the dynamic problem, based on FORM principles, again defining the contour as a locus of all possible FORM design points. We apply these contours to study a practical load combination problem involving the vertical bending moment of a ship. Finally, we present some data on the accuracy of several methods of analysis, and the effect of modeling choices for the input variables. The FORM method is shown to be practical and quite accurate for a variety of situations. Keywords: Reliability, FORM, dynamic, design contours. 1

Introduction

In structural reliability, it is common to relate the failure probability p f to a limit state (or failure surface) g(X) involving a vector X of random variables. The probability distribution of X is presumed to be known and failure is presumed to occur if g(X) < 0 . In many practical situations the random quantities of interest relate to the demand D on the structure, while the capacity C is determined by the designer. In this situation C is treated as deterministic and the limit state condition can be written as g(X) = C − D(X) . *

Corresponding author. Tel. 979-830-7131. E-mail: [email protected]

1

This study will be written using this simplified formulation. In cases of dynamic responses to a load process X(t) , we commonly use a Poisson model to estimate the probability of failure, p f (T) , during a specific time interval T. This only requires knowledge of the mean out-crossing rate by X(t) into the unsafe region, denoted as ν D[ X (t )] (C) . Standard FORM (First-Order Reliability) methods can be used to approximate p f for given probability distributions of X for either the static or dynamic situation (e.g., Madsen et al.,1986). First the Rosenblatt transformation is used to map X into a vector U of independent standard Gaussian random components. This maps the failure condition of D(X) = C into a corresponding U-space surface Dˆ (U) = C . In either the static or the dynamic problem, interest is focused on the most likely point on this limit-state surface. In the standard Gaussian U-space this point u * is simply the point on the limit state that is nearest the origin. The FORM analysis then approximates the limit-state surface by its tangent plane at the point u *, thus simplifying the calculation of the probability of failure. In the static case the FORM approximation of p f depends only on the distance of u * from the origin, denoted as β (u*) , and in the dynamic case it depends on crossing rate information. The design problem is the inverse of the above procedure. That is, there is now a given value of p f but at least some aspects of the capacity C are unknown, since they depend on design parameters that have not yet been chosen. In the static case, inverse FORM methods have been suggested (e.g., [8] and [1]). The resulting "design contours" may be viewed as the locus, in either U-space or X-space, of all possible design points that produce the desired p f . These contours are simply spheres in U-space for the static problem, since p f for any point u depends only on the distance from the origin, β (u) . These contours have been used for a variety of design problems, including environmental issues for offshore structures [9, 2, 4] and loads on wind turbines [7]. The concept of using inverse design contours for dynamic problems was introduced by Leira [5], using a simplification of considering the contours to be ellipsoidal in shape. We seek here to complement this work by developing alternative design contours that meet the conditions of the usual FORM approximations. Thus, we extend the inverse FORM method to dynamic problems. Specifically, we develop contours by using FORM principles to identify the locus of all possible design points having the crossing rate that gives the specified probability of failure. Since the FORM estimate of the crossing rate depends not only on β (u) but also on given information about the crossing rates of X, the U-space contours are generally not spherical. After developing these ideas in a general matrix context, we apply the method to study a two-dimensional load combination problem. We compare the FORM results with those of Leira's ellipsoidal method and with direct numerical integration that does not include the simplifying assumptions of FORM. We do this for several different choices for the characteristics of the loading on a ship, and also for parameter values that extend well beyond reasonable values for the ship bending example. Finally, we introduce an alternate model to examine the effect of an assumption about crossing rates as used in the FORM method.

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2 Mathematical Formulation The equations for the probability of failure, as described above, are Static case: p f = P[D(X) > C]

(1)

(

)

Dynamic case: p f = 1 − exp −T ⋅ ν D[ X (t )] (C)

(2)

in which the dynamic result is based on the Poisson approximation for the probability of a crossing by the stationary time history X(t) into the failure region during a time interval of duration T. We will write the Rosenblatt transformation and its inverse as (3) U = f(X) X = h(U) This mapping of X into the standard Gaussian U, also maps the limit state D(x) = C into Dˆ (u) ≡ D[h(u)] = C . The design point (or "characteristic point") u * is the most likely point on this limit-state surface, which means it gives the minimum value of β (u) ≡| u | for all u with Dˆ (u) = C . The FORM approximation then replaces the limit-state surface in U-space with the plane that is tangent to the surface at u * . With this approximation, the failure condition of Dˆ (u) > C can be written as Z(u) > β (u*) in which Z is a unit-variance scalar Gaussian random variable that is a linear function of u. One way to think of Z is as the projection of the vector from the origin to u onto the vector from the origin to u * . For this standard Gaussian measure of demand, Eq. (1) gives Static case: p f = Φ[−β (u*)] (4) in which Φ denotes the cumulative distribution of the standard Gaussian random variable. Similarly the dynamic probability of failure is based on the rate of crossing of the FORM plane by U(t) . This is the same as crossings by the unit-variance Gaussian time history Z(t) ≡ Z[U(t)] . Thus

(

Dynamic case: p f = 1 − exp −ν Z (t ) [β (u*)]T

)

(5)

Finding the crossing rate of Z(t) in the Gaussian U-space requires knowledge of the joint distribution of Z(t) and Z˙ (t) . Some caution is needed at this point. The obvious choice is to say that Z(t) is a Gaussian process, for which Z(t) and Z˙ (t) are jointly Gaussian. We will use this assumption in the following FORM analysis, but it is not directly implied by the Rosenblatt transformation in Eq. (3). All that is assured by the Rosenblatt transformation is that U(t) is a standard Gaussian random variable for any value of t. It does not assure that U(t1 ) and U(t 2 ) are jointly Gaussian for t1 ≠ t 2 , which ˙ (t) is Gaussian, and jointly Gaussian with U(t) --that is required in order to assure that U is to have U(t) be a standard Gaussian process. The special case in which U(t) is a standard Gaussian process requires that X(t) is a translation process [3], which means that Eq. (3) gives a standard Gaussian process U(t) . An alternate assumption for the ˙ (t) will be presented in Section 5. distribution of X For a standard Gaussian process we know that

3

2

(6) ν (u) = ν 0e − u / 2 in which ν 0 is the mean crossing rate of zero. Thus, using the translation process assumption in Eq. (5) gives 2 Dynamic case: p f = 1 − exp⎛⎝ −ν Z (t ) (0) T e − β (u*) / 2 ⎞⎠ (7) The design problem is the inverse of the above procedure. That is, there is now a given value of p f and the problem is to find the design point x* out of all the x values that give this value of p f . We will consider p f contours in X-space or U-space that contain all points with the specified value of p f . The problem is to find the point u * that not only is on the p f -contour but also is the most likely point on a limit-state surface passing through that point. Since the value of the capacity C is not known a priori in the design problem, neither is the limit state. Thus, one can picture a multi-dimensional limit-state map in either Xspace or U-space with each limit-state surface corresponding to a different value of C. One such surface will pass through each u point. Since u* is the most likely point on the appropriate p f -contour, it is necessary that the limit-state surface is locally orthogonal to the vector from the origin to u*. This orthogonality condition identifies u*, and this point is then mapped back into the X-space to give the so-called "design point" as x* = h(u*) and the design value of the capacity C. 2.1 Static inverse FORM The demand quantity in the inverse problem may be of various forms, such as a load, a load effect, or a structural response. The object is to find a design value of this quantity based on a given probability of failure p f . As in Eq. (4) we see that all points in U-space having a probability of failure p f are on a sphere with radius

β (u) = −Φ −1 ( p f )

(8)

Each of these points can also be identified in Cartesian coordinates by using direction cosines:

u j = β (u)cos(θ j )

(9)

in which θ j is the angle between the u j axis and the vector from the origin to u. One can now envision a family of limit-state surfaces in the U space. Each of these surfaces represents a collection of points satisfying C = Dˆ (u) for a particular value of C. In particular, there will be one such surface passing through every point on the constantβ contour. Since the static p f -contour is spherical, the condition of local orthogonality between the vector from the origin to u* and the limit-state surface makes it clear that u* must be a point of tangency between the p f -contour surface and the appropriate limitstate surface. This point must give the largest value of the demand on the p f -contour. Thus, to find the controlling design situation for this value of p f we only need to search 4

the contour for the point C = Dˆ (u) with the largest value of Dˆ (u) . If the system is designed with a capacity equal to this demand, then the design will be deemed adequate for all other demands with the same value of p f . Using the Rosenblatt transformation in (3), the u* design point and/or the entire p f -contour surface can be mapped directly into X-space. 2.2 Dynamic inverse FORM It has become customary to define the “characteristic” (or most probable) dynamic response level as the level with a mean recurrence time of T. The Poisson estimate of first-crossing rate then gives p f = 1 − e −1 a s t h e probability of failure during the time interval T in Eq. (2). From Eqs. (2) and (6) we find the characteristic value

β cr = [2ln(N)]1/ 2

(10)

in which N = ν 0T is the average number of response cycles over the duration T of interest. We now consider a demand process Z(t) that is a linear combination of the independent Ui (t) processes:

Z(t) = ∑ Ui (t) cos(θ i ) = ∑ Ui (t) i

i

ui β (u)

(11)

in which the final form is based on Eq. (9). Because the U j (t) are all standard Gaussian, any U-space p f - contour that could contain the characteristic extreme must approach the limiting value of β j = [2ln(N j )]1/ 2 on the u j axis--that is when uk = 0 for all k ≠ j . Here N j is the average number of cycles of the jth component over the duration T of interest. What remains in question, of course, is how the contour should be defined in intermediate cases. Leira [ 5 ] h a s suggested an ellipsoidal contour of

∑ (u j / β j )2 = 1

β j = [2ln(N j )]1/ 2

(12)

j

The following sections provide an alternate approach. The formalism of FORM seeks results that are exact within the limitations of a linearized limit-state surface in U-space and the Poisson approximation of the crossing rate. Thus, the starting point is again the demand quantity Z(t) in Eq. (11). Since we consider the U j (t) components to be standard Gaussian processes, Z(t) is also a standard Gaussian process. This implies that we can again use the standard result from Eq. (10) for its characteristic value: zcr = [2ln(N Z )]1/ 2

(13)

in which N Z now represents the average number of cycles of the combined response Z(t) during the time interval T. It only remains to estimate N Z . 5

The general load combination problem, which we seek to address here, includes multiple time-varying components, X i (t) , with each having its own frequency content. In order to estimate N Z it is convenient to note that for a standard Gaussian process, such as Z(t) , the zero-crossing rate ν Z 0 is proportional to the standard deviation σ Z˙ of the derivative process:

σ Z˙ = ω Z 0σ Z = 2πν Z 0σ Z 0 = 2πν Z 0

(14)

Here ν Z 0 = ω Z 0 /(2π ) is the mean frequency in Hz. To use Eq. (14), we differentiate (11): Z˙ (t) = ∑ U˙ j (t)cos(θ j )

(15)

j

The variances then satisfy

σ Z2˙ = ∑ σ U2˙ cos2 (θ j ) j

(16)

j

and Eqs. (14) and (16) give ⎛ ⎛ Tσ ˙ ⎞ 2 ⎞ σ Z˙ Uj T 2 ⎜∑⎜ N Z = ν Z 0T = T= ⎟ cos (θ j )⎟ 2π 2π ⎜ j ⎝ 2π ⎠ ⎟ ⎝ ⎠

1/ 2

⎛ ⎞ = ⎜ ∑ N 2j cos2 (θ j )⎟ ⎝ j ⎠

1/ 2

(17)

in which N j is the average number of cycles of U j (t) or X j (t) during the time interval T. We can use Eqs. (17), (13) and (8) with β (θ ) = zcr as a recipe to generate a U-space p f -contour surface by sweeping over all possible θ j values, similar to the parametric form in Eqs. (10) and (11). It is clear that the proper limiting values of β j = [2ln(N j )]1/ 2 are found on each u j axis, thus agreeing with Eq. (12) at these particular points. Note that the p f -contour surface in U-space depends only on the N j and θ j values. In particular, it does not depend on the Rosenblatt transformation, so it does not depend on the probability distributions of the X j (t) components. There are at least two possible ways to use the dynamic FORM p f -contour surface: 1) as stated earlier, the true FORM solution is to find the point u* on the contour such that the vector u* is orthogonal to the design limit-state surface passing through that point, and 2) a cruder, but conservative, approximation is to find the largest demand value anywhere on the contour. Note that these two approaches are the same for the static problem in which the contours are spherical, but they generally give different results in dynamic problems. This will be investigated in numerical examples.

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3

Numerical Example

We will limit our numerical investigation to a problem with a two-dimensional X-space and U-space and the investigation will be limited to non-negative values of U1 and U 2 . There are at least two reasons for this choice. The first reason is that some of the calculations are significantly simplified, and the second reason is that meaningful results can be shown graphically in two-dimensional space. In addition, it may be noted that there are non-trivial problems that do fall into this category. 3.1 Two-dimensional simplifications of dynamic inverse FORM One simplification in two dimensions is that only one angle is needed to define the location of a point. In particular, we let θ denote θ 1 in the preceding equations, so that cos(θ 2 ) = sin(θ ) , giving the description of any point u as (18) u = β (u)[cos(θ ), sin(θ )] and the demand term in Eq. (11) as (19) Z(t) = U1 (t)cos(θ ) + U 2 (t)sin(θ ) Also, Eqs. (9) and (17) give the number of cycles of Z(t) during the T interval as 1/ 2

⎛ N 2 u 2 + N 22 u22 ⎞ (20) = ⎜ 1 12 ⎟ ⎝ u1 + u22 ⎠ Note that the planar FORM approximation of the p f -contour surface in U-space now becomes a straight line. As a further simplification we will consider the demand to depend on a process Y(t) that is the sum of two independent random processes: (21) Y (t) = X 1 (t) + X 2 (t) so that the limit state for any given value of Y is a straight line with a slope of –45° in Xspace. Of course, the Rosenblatt transformation will generally produce a curved limit state in U-space and a curved p f -contour in X-space. The only exception is when X 1 (t) and X 2 (t) are both Gaussian, so that the Rosenblatt transformation is linear. We know that the critical point u* = (u1 *, u 2 *) in U-space must correspond to a vector that is orthogonal to the limit-state curve at that point. One can estimate this point geometrically directly from a contour plot, but a more accurate evaluation can be calculated analytically by considering the slope of the limit state at any point in U-space. In particular, we note that an incremental movement in X-space with slope δx 2 /δx1 corresponds to a movement in U-space with slope N Z = [N12 cos 2 (θ ) +

N 22 sin 2 (θ )]1/ 2

δ u2 f 2 '(x 2 )δx 2 h1 '(u1 )δx 2 = = δ u1 f1 '(x1 )δx1 h2 '(u2 )δx1

(22)

A constant-y line in X-space has δx 2 = −δx1 so that the slope of the U-space limit state at point u * must be −h1 '(u1*) /h2 '(u2 *). Thus, the orthogonal vector u * must have a slope of h2 '(u2 *) /h1 '(u1*) , and the numerical condition for the critical point can be written as u2 * h2 '(u2 *) = u1 * h1 '(u1*)

(23)

7

For this simplified two-dimensional problem it is also feasible to obtain numerical results without using the FORM approximation of the limit state in U-space. We will use the linear limit state Y=y in X-space along with the Poisson approximation for the time of first crossing of this line. The rate of crossings can be written directly as a triple integral over the possible values of the components [X 1 (t), X˙ 1 (t), X˙ 2 (t)] (see Appendix). In this ˙ (t) also involves the integral, of course, the joint probability density of X(t) and X component X 2 (t) = y − X 1 (t) : ⎛ p X X X˙ X˙ (x1, y − x1, x˙ 1, x˙ 2 ) ×⎞ (24) NY (y) = T ∫ ∫ ∫ ⎜ 1 2 1 2 ⎟ dx1dx˙ 1dx˙ 2 ( x˙ 1 + x˙ 2 )H( x˙ 1 + x˙ 2 ) ⎠ ⎝ in which the probability density function is the normalized Gaussian form. One way to simplify this rather cumbersome calculation is to use the Rosenblatt mapping into Uspace and the condition that U1 (t) and U2 (t) are independent Gaussian processes. This allows simplification to a single integral of 2

2

1/ 2 1 e − (u1 + u 2 ) / 2 2 NY (y) = N1 [h1 '(u1 )]2 + N 22 [h2 '(u2 )]2 ] du1 (25) 1/ 2 ∫ h '(u ) (2π ) 2 2 in which u2 = f 2 [y − h1 (u1 )] and the fi (xi ) and hi (ui ) functions come from the Rosenblatt mapping. Numerical integration is generally required for evaluation of this final integral. This procedure involving numerical integration is quite inefficient compared to FORM, since the numerical integration must be performed repeatedly as one seeks the y value giving NY (y) =1 . The method is introduced here only to investigate the accuracy of the FORM approximation for some selected systems.

(

)

3.2 Ship bending moment example The physical example that we will consider here is based on the one investigated by Leira [5]. It involves the combined vertical midspan bending moment response, Y (t) in Eq. (21) in which X 1 (t) is the still-water bending moment and X 2 (t) is the waveinduced bending moment. It is further assumed that X 1 (t) and X 2 (t) have average cycle numbers of N1 = 120 and N 2 = 10,000 over the duration T of interest. The Rosenblatt transformation then gives N1 and N 2 as the average number of cycles of the Gaussian U1 (t) and U 2 (t) processes. Note that the N j cycle count must be the rate of up-crossings of the median of X j (t) . As noted before, the p f -contour in U-space will be the same for any choices of the probability distributions of the X j (t) components. Figure 1 shows this FORM contour, along with the elliptical approximation used in [5]. Both the contours show the appropriate values of β1 = [2ln(N1 )]1/ 2 = 3.09 and β 2 = [2ln(N 2 )]1/ 2 = 4.29 for the limiting cases of θ = 0 and θ = π /2 , respectively. For intermediate angles, however, the FORM-based contour consistently yields larger values of the radius β . The differences are especially pronounced when the "slow" process U1 (t) dominates. When a small contribution of the "fast" process U 2 (t) is introduced (i.e., θ slightly greater than 0) the cycle number N Z of the combined demand increases quickly, an effect that is not included in the elliptical contour. 8

The plot also highlights the difference mentioned earlier between the two methods of using the FORM-based contour. The true (FORM-predicted) demand may or may not be the maximum demand anywhere on the contour. For greatest accuracy, the FORM-based contour need not be searched over its entirety, but rather evaluated at the specific angle θ that satisfies the orthogonality condition of Eq. (23) for the problem at hand. In some practical cases, however, the maximum demand on the FORM contour is a close approximation of the true FORM design point. This is shown in the examples that follow. 3.2.1 Case 1: Gaussian X 1 and X 2 We will now consider X 1 (t) and X 2 (t) to be independent Gaussian process, and further simplify the problem by considering these two inputs to be mean-zero. After finding the characteristic point or contour in X-space for this simplified problem we can simply add a non-zero mean mi to each xi value in the solution to obtain answers including mean values. Using this mean-zero approach we can characterize each X i input by only two parameters--the standard deviation σ i and the average number of cycles Ni . The first situation we will study has identical input probability densities of 2

2

p X1 (x1 ) = e − x1 /(2σ 1 ) /[(2π )1/ 2 σ 1 ]

σ 1 = 500

(26) 2 2 p X 2 (x 2 ) = e − x 2 /(2σ 2 ) /[(2π )1/ 2 σ 2 ] σ 2 = 500 with the units in X-space being MN-m. The Rosenblatt transformations are now simply (27) ui = fi (xi ) = xi / σ i xi = hi (ui ) = σ i ui Thus hi '(ui ) = σ i and Eq. (23) gives the mandatory condition for the critical point as u2 * = u1*. The point on the p f -contour of Fig. 1 that satisfies this condition has

θ * = 45° and β * = 4.21, giving u2 * = u1* = 2.98 . The mapping into X-space gives x 2 * = x1* = 1,490 and y cr = x1 * + x 2 * = 2,980 . This example was studied by Leira [5], but with mean values of X 1 (t) and X 2 (t) of m1 = 1,000 and m2 = 500 . Adding these values would give the FORM solution for Leira's problem as x* = (2,490, 1,990) and y cr = 4,480 . It may be noted that there is no real need for the Rosenblatt transformation for this totally Gaussian problem since the FORM solution can be obtained directly in X-space. In particular, Y = X 1 + X 2 is also Gaussian, and its standard deviation is (28) σ Y = (σ 12 + σ 22 )1/ 2 In a procedure analogous to Eq. (10), but now rescaled to reflect the standard deviation of Y, we can say that the characteristic value of y is (29) y cr = σ Y [2ln(NY )]1/ 2 and N y is given, as in Eq. (20), as (N12σ 12 + N 22σ 22 )1/ 2 (30) (σ 12 + σ 22 )1/ 2 Thus, the design point, or characteristic value, of Y can be evaluated directly from Eqs. (29) and (30). The results are identical to those obtained from the U-space analysis. The reason for including this totally Gaussian problem in this investigation is to confirm the validity of the U-space FORM method. The FORM-based contour results are NY =

9

in complete agreement with the direct solution, since the Rosenblatt transformations are linear so that the limit state in U-space is linear without any FORM approximation. The only approximation in the calculation is the use of the Poisson formula for the crossing rate into the failure condition. For this Gaussian X(t) problem, both the direct solution and the FORM method are exact within the limitations of the Poisson approximation. Figures 2 and 3 show the p f -contours and design limit states in both U-space and Xspace. Recall that using the maximum Y value on the p f -contour is generally more conservative than using the FORM orthogonality condition of (23). For this example, though, the results are so similar as to be indistinguishable. The plots also include the elliptical U-space p f -contour from Eq. (12). The maximum Y value on this contour is also shown as an estimate of y cr . Its value of 2,650 is seen to be significantly smaller than the FORM value of 2,980. This 11% discrepancy suggests that the elliptical result is non-conservative for this problem. Even though the FORM results are known to be exact for this totally Gaussian problem, the numerical integration procedure of Eq. (25) has also been investigated. The result was identical to the FORM value of 2,980. We have also investigated this Gaussian-densities problem with the intensities of the two loadings changed such that the response is dominated by the low-frequency stillwater term. In particular, we now have (31) σ 1 = 1500 σ 2 = 250 As before hi '(ui ) = σ i and Eq. (23) gives the mandatory condition for the critical point as u2 * = u1* /6 . The point on the p f -contour of Fig. 1 that satisfies this condition has

θ * = 9.46° and β * = 3.85 , giving u* = (3.80, 0.633) . The mapping into X-space gives x* = (5,700,158) and y cr = 5,860 . The maximum Y value on the FORM contour now gives the conservative value of 6,030 for y cr , the elliptical contour gives a value of 4,760. The numerical integration of Eq. (25) gave a value about 2% below the exact FORM result, indicating that the numerical integration procedure sometimes introduces small errors. Note that the result from the elliptical contour is now about 18% below the FORM result. Figures 4 and 5 show the p f -contours and design limit for this situation dominated by the low-frequency component. As noted before, adding only a small highfrequency loading can significantly increase the number of cycles in the combined loading in this low-frequency dominated situation. 3.2.2 Case 2: Gaussian X 1 and exponential X 2 We will use the same standard deviation values as were used in the prior investigations. It should be noted, though, that we are not shifting the exponential distribution of X 2 to be mean-zero. Thus, the first situation has

p X1 (x1 ) = e − x1 /(2σ 1 ) /[(2π )1/ 2 σ 1 ] 2

− x 2 /σ 2

2

σ 1 = 500

σ 2 = 500 The Rosenblatt transformation for X 1 is unchanged from Eq. (27): f1 (x1 ) = x1 / σ 1 , h1 (u1 ) = σ 1u1 and the transformation for X 2 is (33) f 2 (x 2 ) = Φ −1 (1 − e − x 2 /σ 2 ) h2 (u2 ) = −σ 2 ln[1 − Φ(u2 )] p X 2 (x 2 ) = e

/ σ 2 for x 2 ≥ 0

(32)

10

Again h1 '(u1 ) = σ 1 and 2

σ2 e− u2 / 2 h2 '(u2 ) = (34) 1 − Φ(u2 ) (2π )1/ 2 The point on the contour satisfying the orthogonality condition in Eq. (23) has θ * = 77.2° and β * = 4.29 , giving u* = (0.951,4.18) . The mapping into X-space gives x* = (475, 5,570) so that y cr = 6,040 . The numerical results are shown in Figs. 6 and 7, in the same form as in the previous section, including the elliptical contour. This situation where the “fast” process dominates ( θ near π /2 ) is one in which the elliptical and FORM contours show the greatest agreement. Thus, the FORM contour value of y cr = 6,040 is just slightly above the value of y cr = 5,940 from the ellipsoidal contour. It is again found that the maximum y value on the FORM contour almost coincides with the FORM design point, and both agree with the value of 6,050 found by numerical integration of Eq. (25). The same procedure has been applied for the dominant still-water situation of Eq. (31) but with the probability density of Eq. (32). The critical point on the p f -contour now has θ * = 13.9° , β * = 3.95 , u* = (3.83, 0.946) . x* = (5,750, 440) , and y cr = 6,190 . The maximum Y value on the FORM contour is 6,350 and the elliptical contour gives a value of 5,050, which is again 18% lower than the FORM result. Numerical integration of Eq. (25) gives 6,330. Comparing these numbers with those in the previous section shows that changing the probability density function of the high-frequency component has little effect in this situation where the loading is dominated by the low-frequency component. The contour plots for this situation are very similar to those in Figs. 4 and 5, so are not presented here. 3.2.3 Case 3: Gaussian X 1 and large-tail X 2 One aspect of testing the dynamic-inverse-FORM method being presented here is to establish its capability of solving problems in which the X 1 and X 2 inputs have significantly different probability distributions. The Gaussian-exponential Case 2 did provide positive evidence that the method worked well when the symmetric Gaussian X 1 was combined with a one-sided X 2 . However, one may also note that the important part of the probability distribution in reliability analysis is typically the tails of the distribution, and both the distributions in Case 2 have tails with a form of exponential decay for very large xi . In order to introduce a more severe variation in distributions, we will now consider the same Gaussian X 1 as before, but combine it with a one-sided X 2 for which the tail of the probability density function only decays like x 2−4 for very large values of x 2 . We will use the same standard deviation values as were used in the prior investigations. Again we are not shifting the one-sided distribution of X 2 to be meanzero. The first situation has

11

2

2

p X1 (x1 ) = e − x1 /(2σ 1 ) /[(2π )1/ 2 σ 1 ]

σ 1 = 500

(35) 3b 3 2σ 2 with b = σ = 500 2 (x 2 + b) 4 31/ 2 The Rosenblatt transformation for X 1 is again given by Eq. (27): f1 (x1 ) = x1 / σ 1 , h1 (u1 ) = σ 1u1 and the transformation for X 2 is ⎛ ⎛ 31/ 2 x ⎞ −3 ⎞ 2σ −1 2 f 2 (x 2 ) = Φ ⎜1 − ⎜1 + (36) h2 (u2 ) = 1/ 22 [1 − Φ(u2 )]−1/ 3 − 1 ⎟ ⎟ ⎜ ⎝ ⎟ 2 σ 3 ⎠ 2 ⎝ ⎠ Again h1 '(u1 ) = σ 1 and p X 2 (x 2 ) =

(

)

2

2σ 2 e− u2 / 2 h2 '(u2 ) = 3 / 2 (37) 3 [1 − Φ(u2 )]4 / 3 (2π )1/ 2 The point on the contour satisfying the orthogonality condition in Eq. (23) now has θ * = 89.3° and β * = 4.29 , giving u* = (0.0512, 4.29) . The mapping into X-space gives x* = (25.6, 27,300) so that y cr = 27,300 . The results from the numerical integration of Eq. (25), the maximum y value on the FORM contour, and the maximum on the elliptical contour are all essentially the same as the full FORM result of 27,300. The numerical results are shown in Figs. 8 and 9, in the same form as in the previous sections. It may be noted that the large-tail probability distribution used for X 2 in this case results in significant changes in the shape of the p f -contour in X-space, as well as increasing the values of X 2 . However, this contour shape has essentially no effect on the design-point solution, since that occurs for θ near π /2 and the shape change is for smaller θ values. Next we consider the effect of the large-tail distribution of X 2 , as given in Eq. (35) for the situation with σ 1 = 1,500 and σ 2 = 250 -- a situation in which the design point has been dominated by the low-frequency still-water component X 1 in the earlier examples. The search for the point on the p f -contour satisfying the orthogonality condition of Eq. (23) yields a result that was not encountered in the earlier examples. In particular, it is found that there are three points on this U-space contour that satisfy Eq. (23): (θ * , β * ) = (5.24°, 3.69), (29.9°, 4.13), and (85.8°, 4.29). The y cr values, calculated from these three points are 5,630, 6,140, and 13,900, respectively. This "multiple solution" phenomenon is clearly due to the unusual shape of the X-space contours for this combination of probability distributions, as noted in Fig. 9. In order to identify an acceptable design point we must use the solution of Eq. (23) that gives the largest value of y cr . Thus, we say that the FORM solution has

(θ * , β * ) = (85.8°, 4.29) , which gives u* = (0.0313, 4.28) , x* = (469, 13,400) , and y cr = 13,900 . Note that the large value of θ * indicates that the design point is not dominated by X 1, even though σ 1 = 6σ 2 . The reason for this must be related to the fact that the design point is most influenced by the tails of the distributions, which is notably large for this distribution of X 2 . It can be anticipated that still larger values of σ 1 / σ 2 would result in domination by X 1 for these probability forms. 12

As in the other situations studied with a large value of θ * , the results of all the different analysis procedures are essentially identical to the full FORM solution No plots are included for this situation, since they are very similar to Figs. 8 and 9. The issue of multiple points satisfying Eq. (23) can be better understood by rewriting that equation as u h '(u ) / σ 2 Q(u1,u2 ) = σ 1 / σ 2 with Q(u1,u2 ) = 1 2 2 (38) u2 h1 '(u1 ) / σ 1 Note that in each of the examples considered, the value of hi ' (ui ) is proportional to σ i so that Q(u1,u2 ) does not depend on the values of σ 1 or σ 2 . This is also true for any other model in which the probability distribution can be written in terms of the single parameter σ i . Figure 10 shows plots of Q(u1,u2 ) versus θ for each of the distributions considered here. The Gaussian-Gaussian and Gaussian-exponential distributions of Secs. 3.2.1 and 3.2.2, respectively, are both seen to give a monotonic Q(u1,u2 ) curve as θ is varied. Thus, there is exactly one θ value for which the curve crosses any level σ 1 / σ 2 . For the large-tail situation of Section 3.2.3, though, it is clear that there are multiple crossings of some levels of σ 1 / σ 2 . In particular, there are two down-crossings and one up-crossing if 4.15 < σ 1 / σ 2 < 19.6 , including this example with σ 1 = 6 σ 2 . Clearly a situation with σ 1 / σ 2 > 19.6 will give a design point that is dominated by the lowfrequency still-water component X 1 3.2.4 Summary of ship-bending problem results The most significant conclusion that can be drawn from the numerical results is that there are situations in which the elliptical contour gives non-conservative results. That is, it can give a design point that does not meet the standard of only one expected crossing during the time interval studied. The other methods investigated give quite accurate results for the particular situations studied. In particular, it seems to be adequate for these particular examples to use either the maximum y value on the FORM contour, or the true FORM solution of finding the point on the contour that satisfies the condition of being orthogonal to the appropriate limit state. Of course, the accuracy of this single example problem with only two choices of parameters does not necessarily indicate that the same accuracy would apply to other problems. This matter will be investigated in somewhat more detail in Section 4. It may be noted that the one-sided elliptical and large-tail distributions used here for X 2 are not mean-zero. In particular, the exponential distributions have a mean value of

µ 2 = σ 2 and the large-tail distributions have µ 2 = b /2 = σ 2 /31/ 2 . One can find the results for these distributions shifted to have zero mean values by simply subtracting µ 2 from each y cr value. That has not been done here, since the µ 2 values are so small compared to the y cr values that removing them has very little on effect on y cr and no significant effect on the percentage errors when comparing methods.

13

4

Extended Numerical Comparisons

The results in Section 3 have demonstrated the usefulness of the inverse FORM contour for a practical dynamic problem and have also shown that the elliptical contour sometimes gives unsatisfactory results. In particular, the numerical results have consistently given the maximum y value on the FORM contour as in quite good agreement with the results obtained by using the full FORM approach based on orthogonality to the limit-state at the design point in U-space. The focus of this section is a search for situations in which either of these positive findings for the FORM contour may come into question. 4.1 Gaussian X1 and Exponential X2 The range of parameters now investigated for the situation with Gaussian X 1 and exponential X 2 is much broader than was considered for the ship-bending example. In particular, σ 1 / σ 2 now varies from 10 −3 to 10 3 and N1 /N 2 varies from 10 −4 to 10 4 . The values of both N1 and N 2 have been restricted to be no smaller than 100. Over this very broad range of situations it has been found that the FORM results are in very good agreement with the numerical integration results of Eq. (25). The discrepancy was usually less than 1%, although there were limited situations for which it was slightly larger. The largest discrepancy observed was about 4%, and occurred for the rather extreme case of σ 1 = σ 2 , N 2 = 1,000N1. Over all, it seems that the inverse FORM method gives very accurate results over a broad range of parameters in these distributions. The results are not so positive when comparing the full FORM method with the simpler choice of seeking the largest y value anywhere on the p f -contour. In particular, it has been found that the difference between the two results can be quite large in situations where one of the components has a very small value of σ and a very large value of N. For example, if σ 2 = σ 1 /1,000 , N 2 = 1,000 N1 then the maximum-Y approach gives a design point about 38% larger than the FORM result. Similarly, σ 1 = σ 2 /1,000 , N1 = 1,000 N 2 gives a 76% discrepancy. Of course, the maximum-Y approach is always conservative, but it is significantly overly conservative in these rather extreme situations. 4.2 Gaussian X1 and Large-Tail X2 As seen in Section 3.2.3, the combination of a Gaussian component and a large-tail component can give behaviors that are quite different than those found for the Gaussian and exponential components. For the parameters used in the ship-bending study in Section 3.2 it was found that the FORM contour gave results in very good agreement with numerical integration, but that has been found not to be true for some other choices of parameters. This can be illustrated by additional numerical studies of the ship-bending problem. In particular, the maximum y value on the FORM contour is 11% smaller than the value obtained by numerical integration when σ 1 / σ 2 = 17 and the FORM solution based on Eq. (23) or (38) is 15% smaller than the value obtained by numerical integration when σ 1 / σ 2 = 19.5 . Note that these particular discrepancies are non-conservative, in that the FORM contour gives smaller design point values than does numerical integration. For this problem (i.e., this choice of N1 and N 2 ) these are the largest nonconservative values found for the FORM-contour results. 14

As in the previous section, the results from full FORM analysis and the maximum y on the FORM contour have been compared with the results of numerical integration for the broad range of parameters of 10 −3 ≤ σ 1 / σ 2 ≤ 10 3 and , with the smaller value of Ni always restricted to not be less than 100 cycles. The results were quite comparable with those from the ship-bending problem. The general shape of the Q(u1,u2 ) curve was similar to that in Fig. 10, indicating the existence of multiple solutions of the orthogonality condition of Eq. (38) for some values of σ 1 / σ 2 . For some σ 1 / σ 2 value associated with the band of multiple solutions of Eq. (38), the design point value from the full FORM analysis was often found to be near 85% of that found by numerical integration. Away from this σ 1 / σ 2 ratio the FORM results agreed very well with numerical integration. The maximum y value on the FORM contour was somewhat closer to the numerical integration result in those situations in which the full FORM analysis was least satisfactory, but the maximum y approach is overly conservative when one of the components has a very small value of σ 1 and a very large value of N1, as in Section 4.1. 5

An Alternate Model

There are important special cases in which one may be interested in non-Gaussian processes that are not translation processes. That is, there are situations in which the ˙ (t) as independent processes, so none of Rosenblatt mapping does not give U(t) and U the results presented so far in this paper are appropriate. For example, there is a non˙ (t) as trivial class of dynamic models for which the response gives the random variable X being Gaussian and independent of the random variable X(t) at the same instant of time. In particular, this occurs if a Gaussian white noise excitation is applied to a vibratory mechanical or structural system with nonlinear stiffness and linear damping. Of course this model is idealized, but it has often been used to approximate physical problems. We ˙ (t) . will give some limited results for this class of problems with independent X(t) and X As in most of this paper, we will limit our attention here to the special case in which the demand is Y (t) = X 1 (t) + X 2 (t) . In addition, we say that [ X˙ 1 (t), X˙ 2 (t)] is a pair of Gaussian random variables, and [X 1 (t), X˙ 1 (t), X 2 (t), X˙ 2 (t)] is a set of independent random variables for any given value of t. The variance expressions for Y˙ (t) is 2 ⎛ 2 ⎞ 2π ⎜ N X1 (x1m ) N X 2 (x 2m ) ⎟ 2 2 2 (39) σY˙ = σ X˙ + σ X˙ = 2 ⎜ 2 + 2 1 2 T ⎝ p X1 (x1m ) p X 2 (x 2m ) ⎟⎠ in which the procedure of Eq. (14) has been used to write the σ X˙ standard deviation i

values in terms of crossings during an interval T. The term xim in Eq. (39) denotes the median value of X i (t) so that Ni is defined in the same way as in the FORM analysis. For the present calculation, though, this is an arbitrary choice and any other value of xi could be used equally well for the Ni and p X i (xi ) terms. Using Eq. (39) in the standard expression for the number of crossings by Y(t) gives

15

1/ 2

⎛ N2 N 22 ⎞ 1 NY (y) = T pY (y) = (2π ) pY (y)⎜ 2 + 2 (40) ⎟ (2π )1/ 2 ⎝ p X1 (x1m ) p X 2 (x 2m ) ⎠ As before, the design point is the y value giving NY (y) = 1 , so that one can find this point by searching the expression in Eq. (40) over a sequence of y values. The only remaining difficulty is in finding the pdf of Y (t) . This is generally quite simple for the situation considered here with only two X i (t) components. In particular, one can always use numerical integration of pY (y) = ∫ p X1 (x1 ) p X 2 (y − x1 ) dx1 (41) if an analytical solution is not available. Another alternative is to use the Rosenblatt mapping to replace this integral with one in U-space. Although this introduction of the Gaussian distribution may seem desirable, the (y − x1 ) argument maps into the nonlinear form of f 2 [y − h1 (u1 )], so that there does not seem to be any real simplification of the integral. For the special case in which X 1 is Gaussian and X 2 is exponential, as in Section 3..2.2, it is relatively easy to analytically evaluate the integral in (41) giving ⎛ (2y − σ 2 /σ ) ⎞ ⎛ (y − σ 2 /σ ) 2 ⎞ 1 1 2 1 2 ⎟Φ⎜ ⎟ (42) pY (y) = exp⎜ − 2 σ2 2σ2 2σ1 ⎝ ⎠ ⎝ ⎠ Although this expression is uncomplicated, its use is not always trivial. For situations with σ 2 N1 . It should be noted that the differences in these calculations do not reflect an error in either model, but rather the effect of model choice for the X i (t) random processes. The translation model sometimes gives somewhat larger values of the design point than does ˙ model. Thus the FORM approach with the translation model is the independent- X ˙ is modeled as being independent of X(t). conservative in a situation in which X

σ Y˙

6

1/ 2

Summary and Conclusions

The inverse FORM method, previously used for static problems, has been extended to apply to dynamic situations. The method is based on the use of probability-of-failure contours to identify design points. It has been shown that the FORM contour sometimes 16

gives results that are significantly more conservative than those obtained by using an elliptical contour. The full FORM approach is not based on choosing the maximumdemand point on the FORM contour, although that is a simple, conservative alternative. Numerical results have shown that the new FORM technique usually agrees very well with inefficient numerical integration. In the worst situation the design point determined by FORM was 15% smaller than by numerical integration, but this only occurred for a rather extreme choice of the probability distribution of one of the inputs. It has also been shown that that maximum-demand on the FORM contour is overly conservative in some extreme situations, but can be quite good for some more practical problems, such as a ship-bending problem investigated here. ˙ (t) A brief investigation has been made of a non-FORM model in which the X process is Gaussian and independent of X(t). For the system studied, numerical results have shown that this independent-derivative model sometimes gives significantly smaller demand values than inverse FORM, so that use of inverse FORM is conservative in situations in which the other model is more accurate. Appendix: Counting Crossings by Numerical Integration Let H(x) be the unit step function and (43) R(t) = H[Y (t) − y] = H[X 1 (t) + X 2 (t) − y] which "steps" back and forth between zero and unity, increasing to unity at each upcrossing of Y (t) = y and decreasing to zero at each down-crossing of Y (t) = y . This gives (44) R˙ (t) = δ [X 1 (t) + X 2 (t) − y][ X˙ 1 (t) + X˙ 2 (t)] This time history of Dirac delta functions is positive infinity at every up-crossing and negative infinity at every down-crossing. Multiplying by H[Y˙ (t)] = H[ X˙ 1 (t) + X˙ 2 (t)] eliminates the negative Dirac delta functions from a new function (45) N˙ (t) = δ [X 1 (t) + X 2 (t) − y][ X˙ 1 (t) + X˙ 2 (t)]H[ X˙ 1 (t) + X˙ 2 (t)] and also the downward steps from its integral: T N(T) = ∫ 0 δ [X 1 (t) + X 2 (t) − y][ X˙ 1 (t) + X˙ 2 (t)]H[ X˙ 1 (t) + X˙ 2 (t)] dt (46) This makes N(t) be the count of the number of up-crossings of level y and the desired quantity in Eq. (24) is NY (y) = E[N(t)] : ⎛p ⎞ (x1, x 2 , x˙ 1, x˙ 2 ) × X X X˙ X˙ ⎟⎟ dx1dx 2 dx˙ 1dx˙ 2 NY (y) = T ∫ ∫ ∫ ∫ ⎜⎜ 1 2 1 2 ⎝δ (x1 + x 2 − y)( x˙ 1 + x˙ 2 )H( x˙ 1 + x˙ 2 ) ⎠ (47) ⎛ p X X X˙ X˙ (x1, y − x1, x˙ 1, x˙ 2 ) × ⎞ ⎟⎟ dx1dx˙ 1dx˙ 2 = T ∫ ∫ ∫ ⎜⎜ 1 2 1 2 ( x˙ 1 + x˙ 2 )H( x˙ 1 + x˙ 2 ) ⎠ ⎝ The integration with respect to x1 is along the entire real line, so long as the probability density function is defined as zero in any region that is impossible for the given nonGaussian processes.

17

References [1] Der Kiureghian, A., Zhang, Y., Li, C-C., Inverse reliability problem. Journal of Engineering Mechanics, ASCE, 120(5), 1994. [2] Forristall, G.Z., Cooper, C.K., Design current profiles using empirical orthogonal function (EOF) and inverse FORM methods. Proceedings of 1997 Offshore Technology Conference, May 1997. [3] Grigoriu, M., Applied Non-Gaussian Processes. PTR Prentice Hall, Englewood Cliffs, NJ, 1995. [4] Haver, S., Winterstein, S.R., Environmental contour lines: A method for estimating long-term extremes by a short-term analysis. Transactions of the Society of Naval Architects and Marine Engineers, 116, 2009. [5] Leira, B.J.. Combination of multiple extreme bending moment components. Proceedings of 31st Intl. Conf. on Offshore Mech. and Arctic Eng., 2012. [6] Madsen, H.O., Krenk, S., Lind N.C., Methods of Structural Safety. Prentice-Hall, Englewood Cliffs, NJ, 1986. [7] Saranyasoontorn, , K., Manuel, L., Efficient models for wind turbine extreme loads using inverse reliability. Journal of Wind Engineering and Industrial Aerodynamics, 92(10), 2004. [8] Winterstein, S.R., Ude, E.C., Cornell, C.A., Bjerager, P., Haver, S., Environmental parameters for extreme response: Inverse FORM with omission factors. Proceedings. ICOSSAR–93, 1993. [9] Winterstein, S.R., Jha, A.K., and Kumar, S., Reliability of floating structures: Extreme response and load factor design. Journal of Waterway, Port, Coastal and Coastal Engineering, 125(4), 1999.

18

Figure Titles Figure 1. Elliptical and FORM-based p f -contours for with average cycles of N1 =120 , N 2 =10,000 . Figure 2. U-space elliptic and FORM contours, limit states of x1 + x 2 = constant , X 1 and X 2 Gaussian, data from Eq. (26). ! ! Figure 3. X-space elliptic and FORM contours, limit states of x1 + x 2 = constant , X 1 and ! X 2 Gaussian, data from Eq. (26). ! ! Figure 4. U-space elliptic and FORM contours, limit states of x1 + x 2 = constant , X 1 and ! X 2 Gaussian, data from Eq. (31). ! ! Figure 5. X-space elliptic and FORM contours, limit states of x1 + x 2 = constant , X 1 and ! X 2 Gaussian, data from Eq. (31). ! ! Figure 6. U-space elliptic and FORM contours, limit states of x1 + x 2 = constant , X 1 ! Gaussian and X 2 exponential, data from Eq. (32). ! ! Figure 7. X-space elliptic and FORM contours, limit states of x1 + x 2 = constant , X 1 ! Gaussian and X 2 exponential, data from Eq. (32). ! ! Figure 8. U-space elliptic and FORM contours, limit states of x1 + x 2 = constant , X 1 and ! X 2 with large tail, data from Eq. (35). ! ! Figure 9. X-space elliptic and FORM contours, limit states of x1 + x 2 = constant , X 1 and ! X 2 with large tail, data from Eq. (35). ! ! Figure 10. Q(u1,u2 ) function for orthogonality condition of Eq. (38). ! ! ! ! !

4

3

2 Ellipse FORM-based

1

0

0

1 2 3 Standard Gaussian u1

Fig. 1

4

Ellipse FORM-based h1(u1)+h2(u2)=2,650 h1(u1)+h2(u2)=2,980 θ=45°

5 4 3 2 1 0

0

1

2 3 4 Standard Gaussian u1

Fig. 2

5

6

2,500 2,000 1,500 Ellipse FORM-based x1+x2=2,650 x1+x2=2,980

1,000 500 0

0

500

1,000

1,500

2,000

Gaussian x1, Still-water bending moment

Fig. 3

Ellipse FORM-based h1(u1)+h2(u2)=4,760 h1(u1)+h2(u2)=5,860 h1(u1)+h2(u2)=6,030 θ=9.46°

5 4 3 2 1 0

0

1

2 3 4 Standard Gaussian u1

Fig. 4

5

6

1,200 1,000 800 600 400 200 0 0

Ellipse FORM-based x1+x2=4,760 x1+x2=5,860 x1+x2=6,300 2,000 4,000 6,000 Gaussian x1, Still-water bending moment

Fig. 5

Ellipse FORM-based h1(u1)+h2(u2)=5,940 h1(u1)+h2(u2)=6,040 θ=77.2°

5 4 3 2 1 0

0

1

2 3 4 Standard Gaussian u1

Fig. 6

5

6

Ellipse FORM-based x1+x2=5,940 x1+x2=6,040

6,000

4,000

2,000

0

0

500

1,000

1,500

2,000

Gaussian x1, Still-water bending moment

Fig. 7

Ellipse FORM-based h1(u1)+h2(u2)=27,300 θ=89.3°

5 4 3 2 1 0

0

1

2 3 4 Standard Gaussian u1

Fig. 8

5

6

25,000 20,000

Ellipse FORM-based x1+x2=27,300

15,000 10,000 5,000 0

0 500 1,000 1,500 2,000 Gaussian x1, Still-water bending moment

Fig. 9

25

Gaussian X1, Gaussian X2 Gaussian X1, Exponential X2 Gaussian X1, Large-tail X2

20 15 10 5 0

0

0.5

1.0 θ , radians

Fig. 10

1.5