A11107 3101^3

NBSIR 84-2921

Structural Reliability Fundamentals

and Their Application to Offshore

Structures

Emil Simiu

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Building Technology

Gaithersburg, MD 20899

and

Charles E. Smith

Technology Assessment and Research Branch

Minerals Management Service

U.S. Department of the Interior

Reston, VA 22091

September 1984

-QC130

,156

Prepared for: inerals Management Service

I.S. Department of the Interior

eston, VA 22091

34-2921

1934

C . 3. „

NBSIR 84-2921

STRUCTURAL RELIABILITY FUNDAMENTALS

AND THEIR APPLICATION TO OFFSHORE

STRUCTURES

Emil Simiu

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Building Technology

Gaithersburg, MD 20899

and

Charles E. Smith

Technology Assessment and Research Branch

Minerals Management Service

U.S. Department of the Interior

Reston, VA 22091

September 1984

Prepared for: Minerals Management Service

U.S. Department of the Interior

Reston, VA 22091

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige. Secretary

NATIONAL BUREAU OF STANDARDS. Ernest Ambler. Director

TABLE OF CONTENTS

Page

1. INTRODUCTION 1

2. NOMINAL FAILURE PROBABILITIES, SAFETY INDICES, AND LOAD AND

RESISTANCE FACTORS 4

3. SUMMARY AND CONCLUSIONS 19

4. REFERENCES 20

iii
LIST OF TABLES

Page

Table 1. Probabilities of Failure of Four Members Corresponding to

Various Distributions of the Variables 14

LIST OF FIGURES

Figure 1. Index 3 for member with random load and deterministic

resistance 23

Figure 2. Index 3 for member with random load and random resistance . 24

Figure 3. Index 3 in space of variables r r , ur 25

Figure 4. Failure boundaries for members I and II 26

Figure 5. Probability density function f p (rr ) 27

iv
1 . INTRODUCTION

The objective of structural reliability is to develop design criteria and

verification procedures aimed at ensuring that structures built according to

specifications will perform acceptably from a safety and serviceability viewpoint. This objective could in principle be achieved by meeting the following

requirement: failure probabilities (i.e., probabilities that structures or members will fail to satisfy certain performance criteria) must be equal to or less than some benchmark values referred to as target failure probabilities . 0

Such an approach would require [2, 13]:

1. The probabilistic description of the loads expected to act on the

structure

.

2. The probabilistic description of the physical properties of the

structure which affect its behavior under loads.

3. The physical description of the limit states, i.e., the states in

which the structure is unserviceable (serviceability limit states) or unsafe (ultimate limit states). Examples of limit states include:

excessive deformations (determined from functional considerations);

excessive accelerations (determined from studies of equipment performance, or from ergonomic studies on user discomfort in structures

experiencing dynamic loads); specified levels of nonstructural damage;

structural collapse.

4. Load-structural response relationships covering the range of responses

from zero up to the limit state being considered.

5. The estimation of the probabilities of occurrence of the various limit

states (i.e., of the failure probabilities), based on the elements listed

in items 1 through 4 above and on the use of appropriate probabilistic

and statistical tools.

6. The specification of maximum acceptable probabilities of occurrence of

the various limit states (i.e., of target failure probabilities).

The performance of a structure would be judged acceptable from a safety or serviceability viewpoint if the differences between the target and the failure

probabilities were either positive (in which case the structure would be

overdesigned) or equal to zero.

a An alternative statement of this requirement is that the reliabilities

corresponding to the various limit states must be equal to or exceed the

respective target reliabilities (reliability being defined as the difference

between unity and the failure probability).

1
The approach just described is seldom applicable in practice. Owing to physical

and probabilistic modeling difficulties and to the absence of sufficient statistical data it seldom is possible to provide confident probabilistic descriptions

of the loads, particularly within the loading range corresponding to ultimate

limit states. Comprehensive probabilistic descriptions of the relevant physical

properties of the structure are frequently not available. In some cases limit

states are difficult to define quantitatively, particularly with regard to

serviceability. Difficulties may also arise in attempting to describe relationships between loading and structural response that involve material and/or

geometric nonlinearities or contributions by nonstructural elements to the

total structural capacity. For many realistic structures (e.g., redundant offshore structures subjected to wind and wave effects) the estimation of failure

probabilities can be analytically unfeasible or computationally prohibitive,

at least in the present state of the art. A further difficulty is due to the

iterative nature of the design process and the consequent need to perform a reliability check after each step of the iterative design. Finally, there are few agreed upon values of target failure probabilities, particularly for

limit states involving loss of life, as opposed to mere economic loss.*

The reliability analyst is therefore forced to accept various compromises. In

practice, in the absence of sufficient data and of proved probabilistic and

physical models, it may be necessary to use conservative models, or models based

at least in part on subjective belief. In addition, definitions of limit states

may have to be adopted on the basis of computational convenience, rather than

on physical grounds. (For example, ultimate limit states are defined in most

cases as the collapse of individual members, rather than as the collapse of the

structure as a whole;** member collapse is in certain cases conventionally defined

as the attainment of the yield stress at the most highly stressed section of

the member, even though this does not usually entail physical collapse.) Finally,

to simplify the computations, various approximations concerning the mechanical

and probabilistic behavior of the structure may have to be used.

Estimates of nominal (or "notional") failure probabilities are thus obtained

that can differ—in certain instances significantly—from the "true" probabilities. However, if there are grounds to believe that the ratios of nominal to

"true" probabilities for two given designs do not differ significantly, the

two designs may be compared from a reliability viewpoint on the basis of the

respective nominal, rather than "true", probabilities. It would, of course,

be desirable to establish target (i.e., maximum acceptable) nominal failure

probabilities. Attempts are made to infer target nominal failure probabilities

For questions pertaining to safety goals for the operation of nuclear power

plants, see reference 1.

In recent years a number of studies concerned with structural systems have

been reported [ 2 ,3 ,4 , 5 ,6 , 7 ]

.

However the practical usefulness of such studies

remains limited, particularly as far as ocean engineering applications are concerned

.

2
from the reliability analysis of exemplary designs, i.e., designs that are regarded by professional consensus as acceptably but not overly safe. Such

inferences are part of the process referred to as safety calibration against

accepted practice.

While there are instances where such a process can be carried out successfully,

difficulties arise in many practical situations. For example, structural reliability calculations suggest that current design practice as embodied in the

American National Standard A58.1 [8] and other building standards and codes

is not risk-consistent. In particular, estimated reliabilities of members

designed in accordance with current practice are considerably lower for members

subjected to dead, live, and wind loads than for members subjected only to dead

and live loads [9, 13, 17], especially when the effect of wind is large compared

to the effect of dead and live loads. Whether these differences are real or only apparent, i.e., due to shortcomings of current structural and reliability

analyses, remains to be established. Thus, it is not possible in the present

state of the art to determine whether it is the lower or the higher estimated

reliabilities that should be adopted as target values.

In spite of both theoretical and practical difficulties, structural reliability

tools can in a number of cases be used to advantage in design and for code

development purposes. The objective of this report is to present a review of

fundamental topics in structural reliability as applied to individual members,

which are potentially applicable to ocean engineering problems. These topics

include: the estimation of failure probabilities; safety indices; and safety

(or load and resistance) factors.

The earliest justification of current design practice with respect to wind

loading was traced by the authors to Fleming's 1913 monograph Wind Stresses

[10], which states: "Maximum wind loading comes seldom and lasts but a short time. The working stresses used for the loading may therefore be

increased by 50% above those used for ordinary live- and dead-loads."

3
2. NOMINAL FAILURE PROBABILITIES, SAFETY INDICES, AND LOAD AND RESISTANCE

FACTORS

The estimation of nominal failure probabilities3 is an essential task of

structural reliability,, Safety indices are, at least in theory, measures of

failure probabilities. The use of load and resistance factors in design

criteria is intended to ensure that the members to which the criteria are applied have acceptable failure probabilities within any specified period of

interest (e.g. one year, or the lifetime of a structure).

Modeling of Loads as Random Processes and Random Variables

Quantities that vary continuously and randomly with time (e.g., the wind speed,

the wind velocity vector, or the wave height at a given location) can be modeled

as random processes. Quantities that are constant in time, (e.g., the dead

weight), or whose variation in time follows a deterministic law, can be modeled

more simply as random variables.

In problems involving combinations of two or more randomly time-dependent loads,

it is in general necessary to estimate failure probabilities by resorting to

models and techniques drawn from the theory of random processes. If the system

being considered depends, in addition to the randomly time-dependent loads,

upon the random variables, Xi , X2 , ... Xm , estimates of failure probabilities

P(failure |X]_, X2 , ... Xm) are obtained which are conditional upon the values

Xi, X2 , ... Xfjj taken on by these variables. The probability of failure of the

structure, Pf , is then estimated by applying the theorem of total probability

as follows:

Pf = / P(failure|x 1 ,x2,...xk )fx1> x2 ,“’Xk( x l> x2»**‘ xk) dx

1 dx2 ••• dxk

where fx^, Xp , ... Xk

( x l> x 2> ••• xk) = joint probability density function of

Xf , X2 , ... Ak . For treatments of load combination problems based on random

process representations see references 11 and 12.

In problems involving only one randomly time-dependent parameter, £(t), the

question of combining time-dependent random processes no longer arises. It is

therefore convenient in applications to use, in lieu of the random process, C(t),

its largest value during the lifetime of the structure, denoted by X^ n ^ . By

substituting the random variable, x( n ), for the random process, ?(t), the treatment of the reliability problem is simplified considerably. The largest

lifetime value, X^ n ), can be characterized probabilistically as follows:

Fx

(n) (») “

( \( 1 ) Cx>! n

a For brevity, nominal failure probabilities are henceforth referred to simply

as failure probabilities.

4
where = extreme value of c(t) during a time interval t^ = T/n, T = lifetime

of structure, n = integer and F = cumulative distribution function of X^ n '.

The cumulative distribution function of X

VAy

, F is referred to as the

parent distribution of x( n ). Equation 2 holds if successive values of X^D

are identically distributed and statistically independent. An application of

equation 2 is presented in the following example.

Example . Let X^^ = U

a denote the largest yearly wind speed at a given

location. Then x( n ) = U denotes the largest wind speed occurring at

that location during an n-year period (equal to the lifetime of the structure) . It is assumed that the largest yearly wind speed, Ua has an Extreme

Value Type I distribution, i.e.,

f

ti (u ) = exp [-exp(--^)] (3)

u a

d o

It can be shown that

M = Ua

- 0.45 ou (4)

a

o - 0.78 0y ( 5

)

a

where Ua and oy = sample mean and sample standard deviation of the

a largest annual wind speed data, Ua

. From Eqs. 2 through 5 it follows

that the probability distribution of the largest lifetime wind speed, U,

is

:

Fy(u) = exp [-exp(- -) ] (6)

an where

Un - U - 0.45 ay (7)

an = 0.78 ay (7a)

U = Ua + 0.78 0y in n (8)

a °U = aU (9)

a

and n = lifetime of structure in years.

5
Finally, we consider the case where the structure is acted upon by two randomly

time-dependent loads with the following properties: (1) their extreme values

have negligible probability of simultaneous occurrence, (2) their most unfavorable combination occurs when one of the loads reaches its largest lifetime

value, while the other has an "ordinary" (also termed "arbitrary-point-in time"),

rather than an extreme, value3 . Since the "arbitrary-point-in-time" loading

can be modeled by an appropriately chosen time-independent probability

distribution [13], the reliability problem can in this case also be reduced

to one involving only random variables.

Failure and Safe Regions, Failure Boundary

Consider a structure or member subjected to a load Q, and let the value of the

loading that induces a certain limit state in the structure (e.g., the yield

stress) be denoted by R. It is assumed that both Q and R are random variables.

The space defined by these variables is referred to as the load space . By

definition, failure occurs for any pair of values, Q, R, satisfying the relation

R - 0 < 0 (10)

Equation 10 defines the failure region in the load space. The survival region,

or safe region , is defined by the relation

R - Q > 0 (11)

The failure boundary , which separates the failure and safe regions, is defined

by the equation

R - 0 = 0 (12)

Relations similar to equations 10, 11, and 12 can be written in

space , defined by the variables Qe , Re, where Qe is the effect,

induced in the structure by the load Q (e.g., a state of stress

and Re is the corresponding capacity (e.g., the yield stress or deformation) . The equation of the failure boundary in the load

the load effect

or state,

or deformation)

,

a specified

effect space is

Re - Qe = 0 (13)

In general, Q and R are functions of random variables , X2 , ... Xn (e.g.,

aerodynamic coefficients, terrain roughness, cross-sectional area, modulus of

a For example, it may be assumed that these properties characterize the wind

load and the live load acting on members of high-rise building frames.

6
elasticity, breaking strength) 3

, i.e.,

0

II 0 X(—

•

X2 > • • • Xn ) (14)

R = R(XX , x2 , ... xn ) (15)

Substitution of equations 14 and 15 into equations 10, 11, and 12 yields the

mapping of the failure region, safe region, and failure boundary onto the

space of the variables , X2 , . .., Xn . The equation of the failure boundary

thus can be written as

g(Xi, X2 , ...» Xn) = 0 (16)

The well-behaved nature of structural mechanics relations generally ensures

that equation 13 is the mapping of equation 12 onto the load effect space.

Equation 16 is thus the mapping onto the space X]_ , X2 , . . • , Xn not only of

equation 12, but of equation 13 as well. Therefore, once it is made clear at

the outset that the problem is formulated in the load, or in the load effect,

space, it is permissible to refer generically to 0 and Ge as "loads” and to

R and Re as "resistances", and to omit the index "e" in equation 13.

The important case is noted where relations between loads and/or resistances

and more fundamental variables of the problem, Xi, X2 , ... Xn , can only be

obtained numerically. In that case, equation 16 cannot, in general, be

written in closed form.

It is useful in various applications to map the failure region, the safe region,

and the failure boundary onto the space of the variables Yj, Y2 , ..., Yr , defined by transformations

Yi = YiCXi, X2 , ..., Xn ) (i = 1, 2, ..., r) (17)

For example, if in equation 16 Xj. = p, and X2 = U, where p = air density

and U = wind speed, a variable representing the dynamic pressure may be defined

by the transformation Y^ = 1/2 pU , and equation 16 may be mapped onto the

space of the variables Y]_, X3 , X4 , ..., Xn . Another example is the frequently

used set of transformations

*1 = £nR (18)

y2 = JcnQ (19)

a These are sometimes referred to as basic variables. We will use here simply

the term "variables", since what constitutes a basic variable is in - . instances a matter of convention. For example, the hourly wind speed

above ground in open terrain, which is regarded in most applications as a basic variable, depends in turn upon various random storm character 1 sti s, such as the difference between atmospheric pressures at the center and t

<• periphery of the storm, the radius of maximum storm winds, and so forth.

7
It follows immediately from equations 12, 18, and 19 that the mapping of the

failure boundary onto the space Yj_, Y2 , is

YX - Y2 = 0 (20)

or

Si nR - £nQ = 0 (21)

The failure boundary is a point, a curve, a surface, or a hypersurface according

to whether the problems at hand is formulated in a space of one, two, three,

or more than three random variables.

General Expression for Estimation of Failure Probability

Let the failure region in the space of reduced variables3

x, , X

2 r r denoted by ft. The probability of failure Pf, can be written as:

x

n » be

r

p f =

Ia

f

L 1 9 Z

9 r r

(xi , Xo , . . . xn ) dxi dxo ... dxn (22)

r r r r r r

where the integrand is the joint probability density function of the reduced

variables

.

In most cases the estimation of failure probabilities by equation 22 is

computationally unwieldy, if not prohibitive, and the use of alternative methods

is attempted instead. Various such methods, whose applicability depends upon

the characteristics of the problem at hand, are described following the

introduction in the next section of the useful notion of safety index.

Safety Indices

The safety index is a statistic which, under certain conditions that will be

illustrated subsequently, can provide a simple and convenient means of assessing

structural reliability.

a The reduced (or standardized variable, x

r , corresponding to a variable, X, is

defined as X - X

where X and o x are the mean and standard deviation of X, respectively ._ The coefficient of variation (c.o.v.) of X is defined as the ratio ax /X

and is denoted by Vy.

8
We consider a failure boundary in the space of a given set of variables, and

denote by S its mapping in the space of the corresponding reduced variables.

The safety index, 3, is defined as the shortest distance in this space between

the origin and the boundary S [I4] a . The point on the boundary S that is

closest to the origin, as well as its mapping in the space of the original

variables, is referred to as the checking point . For any given structural

problem, the numerical value of the safety index depends upon the set of variables in which the problem is formulated. The examples that follow illustrate

the meaning of the safety index and the dependence of its numerical value upon

the set of variables being used.

Example 1 . It is assumed that the only random variable of the problem is

the load (effect) Q. The resistance—a deterministic quantity—is denoted

by R. The mapping of the failure boundary

0

- R = 0 (23)

onto the space of the reduced variable q r = (Q - Q)/oq (i.e., onto the

axis 0q r —see figure 1) is a point, q , whose distance from the origin

0 is 3 = (R - Q)/<Jq. The safety index represents in this case the difference between the values R and Q measured in terms of standard deviations,

oq. It is clear that the larger the safety index 3 (i.e., the larger

the difference R - Q for any given Oq, or the smaller Oq for any given

difference R - Q) , the smaller the probability that Q > R.

Example 2 . Consider the failure boundary in the load space (equation

10), and assume that both R and Q are random variables. The mapping

of equation 12 onto the space of the reduced variables q r = (Q - 0) oq

and r r = (R - R)/oq is the line [13]

0Q<lr + Q

" GRr r“ R = 0 (24)

(figure 2). The distance between the origin and this line is

3 = R - Q

+ a 1/2

(25)

a This definition is applicable to statistically independent variables. If the variables of the problem are correlated, they can be transformed by a linear operator into a set of independent variables [14], Note that an alternative, generalized safety index was proposed in reference 15, whcse

performance is superior in situations where the failure boundaries 3re r. (see also Chapter 9 of reference 16).

9
This definition of the safety index was suggested in [20]

Example 3 . Instead of operating in the load space R, Q, we consider

equation 20, i.e., the failure boundary in the space , Y£ , defined by

equations 18 and 19. Following exactly the same steps as in the preceding example, but applying them to the variables Yj and Y2 , the safety

index is in this case

3 =

) 172

(26)

Expansion in a Taylor series yields - 1 , - 9

1 Yl = Sin R + (R -R) — - ± (R - R) z —+ ... R

2 R

2

and a similar expression for Y£. It then follows that if R and Q are uncorrelated ,

(27)

Jin R - _L V

2 - (£ n Q - _L V

2

) 3 « (V2 + v

2

) 1 / 2 R 0

(28)

where Vr and Vq denote the coefficient of variation of R and 0, respectively,

If higher order terms in the numerator of equation 28 are neglected, which

is reasonable provided that Vr and Vq are less than 0.3, say,

3 * An(R/Q)

(V? + v

2

) 1/2

'r

1 ’ey

This definition of the safety index was used, e.g., in [22].

(29)

Example 4 . Consider a linearly elastic member whose stresses, 0, can be

written as

0 = aU2 (30)

where a = deterministic influence coefficient and U = wind speed. We assume that a = 0.00267 ksi/(mph) 2

; the mean and standard deviation of

the largest annual wind speed are Ua = 43.73 mph and ajj = 8.61 mph; the mean and standard deviation of the resistance are R = 35.3 ksi and or = 3.39 ksi; and that the lifetime of the member is n = 50 years. From

equations 8 and 9, it follows that the mean and standard deviation of the

10
largest lifetime wind speed are U = 70 mph, ay = 8.61 mph. An expansion

of equation 30 in a Taylor series about the mean, U, yields

0 = aU2 (l + Vy) (31)

and

Vq « 2Vy (32)

i.e., 0 * 13.3 ksi and Oq - 3.27 ksi. The equation of the failure

surface in the space of the variables U, R is

aU2 - R = 0 (33)

and its mapping in the space of the reduced variables ur , r r is

(ur + ^-) 2 =-^- (r r + *-) (34)

°U ao

2 aR

The value of the safety jndex is 0 =^4.31 (figure 3). The coordinates of

the checking point are r

r = -2.51, ur = 3.50, to which there correspond in the U, R space the coordinates U* = 100.14 mph, R* = 26.76 ksi.

It can be verified that the values of the safety index corresponding to

the variables 0, R (equation 25) and £nQ, &nR (equation 29) are

0 - 4.66 and 0 - 3.69, respectively.

Note that the mean and standard deviation of the largest lifetime wind speed or

of the largest lifetime load, which are needed for the calculation of the safety

index, cannot be estimated directly from measured data, but must be obtained

from the probability distribution of the lifetime extreme. This distribution

is estimated from the parent distribution that best fits the measured annual

extreme data. Knowledge of, or an assumption concerning, the parent probability distribution is required for the estimation of the safety index in all cases involving a random variable that represents a lifetime extreme.

Safety Indices and Failure Probabilities: The Case of Normal Variables

Consider the space of the variables Q and R, and assume that both variables are normally distributed. Note that the failure boundary (equation 12) is linear. Since the variate R - Q is normally distributed, the probability of

failure can be written as

P f = F(R - Q < 0)

/2Fgr_

q

/ exp [-*(— -

2

A ) 2

] dx

—00 4 aR-Q

= 1 - $ (_JL_=_n_)

/a

2 + a2 R 0

(35)

11
where $ = standardized normal cumulative distribution function, and the quantity

between parentheses is the safety index, 3, corresponding to the space of the

variables R, Q (equation 25).

More generally, it can similarly be shown that the relationship

P f = 1 - $(0) (36)

is valid if all the independent variables, Xj, X2 , ... Xn are normally distributed, the failure boundary (equation 16) is a linear function, and 3 is the

safety index in the space of the reduced variables x^ , X2 ,

. . . , xn . r r r

Equation 36, of which equation 35 is a particular case, can be used even

if the variables Xj , X2 , . .., Xn are not normally distributed and the failure

boundary g(Xj, X2 , ..., Xn ) = 0 is nonlinear, provided that a transformation of

variables Yi = f^X^) (i = 1, 2, . .., n) can be found such that are normally

distributed and the failure boundary in the space Y^ , Y2 , ..., Yn

is linear.

For example, assume that R and 0 have lognormal distributions. Then Yj = In

Q

and Y2 = &nR are normally distributed, and the equation of the failure boundary

is Yj - Y2 = 0 (equation 20). Applying equation 36 to the variables Yj

and Y2 ,

P f = 1

« 1

£nR - &nQ

SZ 2 , _ 2

°£nR + a£nQ

&n(R/"0)

V 2 + V

2 R Q

(37)

The quantities between parentheses in equation 37 may be recognized,

respectively, as the exact and approximate expression for the safety index, 3,

corresponding to the space of the variables £nR, JlnQ (equations 26 and 29).

Safety Indices and Failure Probabilities: The Case of Non-normal Variables

If the variables Xj, X 2 , ... Xn , are non-normal, or functions thereof are nonlinear, equation 36 is not applicable. The fact that equation 36 does not hold means that members having the same safety index will, in general, have different

failure probabilities. To illustrate the relationship between safety index

and failure probability we consider the four members for which the means and

standard deviations of the load and resistance are listed in table 1. Members

I and II have the same value of 3 in the space of the reduced variables y]_ , y2 (corresponding to equations 18 and 19); their failure boundaries in

that space are shown in figure 4. Members II, III, and IV have the same value

of 3 in the space of the reduced variables r r , q r , representing the resistance

and the load, respectively. The probabilities of failure of the four members

based on the assumption that all the variables are normal are shown in column

8, Table 1. Those corresponding to the assumption that all the variables are

12
Table

I.

Probabilities

of

Failure

of

Four

Members

Corresponding

to

Various

Distributions

of

the

Variables

rsi

hux

•os

i-oc-~

T3Xo

-ooXX

scX

rHuX-u=XXXo

•o

CiX

si

JZ

m

Siu

•j i

c. c ^ c

4J

C

c. z £ — X > XX oe c c

•X

X

XX

X

X

sc _x u

13
lognormally distributed are listed in column 9. Column 10 shows the failure

probabilities based on the assumptions that (a) the load is given by equation

30, (b) the distribution of the wind speed is Extreme Value Type I, and (c)

the reduced variable representing the resistance has the probability density

function f

p (r

r ), shown in figure 5 a

. (From equations 6 and 22 it follows

that the failure probability based on these assumptions has the expression

Pf

-

1

, . u r + 0.45 u r + 0.45,

0.78 lvjxp he*Pt- —,78 1 - - 0.78 |dUf

°u

a°n

2

fu r + JL) 2 -i

On CJtj Or, X / _ fp(rr )dr r (38)

- JL

°R

in which U and Vjj are related to Q and Vq as shown by equations 31 and 32.)

The probabilities of column 10 were obtained by evaluating the integrals of

equation 38.

It is seen from table 1 that to equal values of the safety index, 8, calculated

by equation 25 (which is based on the assumption that the probability distributions of 0 and R are normal), there correspond failure probabilities calculated

by quadrature (column 10) that vary by as much as one order of magnitude (members

II, III, and IV). On the other hand, results of numerical studies reported in

reference 17 showed that, for values of 0 comparable to these of column 9,

equation 38 can be fitted to within a factor smaller than two by the curve

P

f exp [ - (. 8 + 0.45 ^ 0.835

1/0.85,

(39)

where 8 is calculated by equation 29; that is, the probability of failure, Pf

,

is, at least approximately, determined uniquely by the safety index 0 calculated

by equation 29, regardless of the relative values of Q, Vq, R, and Vr. This

interesting conclusion - which, as was just pointed out, does not hold for

the safety index, 8, calculated by equation 25 - is explained by the approximate

similarity between the shapes of the lognormal distribution on the one hand,

and the distributions used in equation 38 on the other hand.

Note, however, that while this conclusion is valid in the particular case just

examined, it does not necessarily hold in other situations. For example, it

a This shape corresponds approximately to published data on the yield stress of

A33 steel, for which the nominal value, the mean value, and the coefficient

of variation are

p. 237].

F

y

= 33 ksi, R * 1.07 Fy , and V R » 0.096 [17; 18,

14
is possible that two members, one subjected to gravity loads and the other to

wind or wave loads, will have widely different failure probabilities even if

their safety indices calculated by equation 29 are nearly equal. In this

case, or in similar cases, a comparative reliability analysis would require

the estimation of the failure probability by equation 22 or by alternative,

approximate methods. A few such methods are briefly described in the following

section

.

Approximate Methods for Estimating Failure Probabilities

We first describe the method referred to as normalization at the checking point

[19]. The principle of the method is to transform the variables, , into a set of approximately equivalent normal variables, x!?, having the following

property:

Px^ (X* ) = f( X;*)

=_J_ <Kx?*) (40)

ox

n r

P

X. (Xi> = F< xi*)

= <D(x. n*)

r (41)

where i = 1, 2, ..., m, the asterisk denotes the checking point, px and Px are the probability density function and cumulative distribution function of X^

,

respectively, f and <}> are the normal and standardized normal probability density

function, respectively, F and $ are the normal and standardized normal cumulative distribution function, respectively, x? is the reduced variable correr sponding to X^, and a x

n is the standard deviation of Xj. From equations

40 and 41 it follows that

ax

n =

4>(*

1

[ P ( X*

) ]

px

.(x*)

X? = xi " $

1 [P

Xi (X*)]a x

n

(42)

(43)

where the bar denotes mean value. Once X? and axn are obtained from

equations 42 and 43, the problem can be restated in the space of the reduced

variables, x^ . The safety index, 3, is the distance in this space between

r the origin and the failure boundary. A computer program for calculating this

15
distance, based on an algorithm proposed in reference 19, is presented in

reference 13. Following the calculation of 3, the probability of failure

is estimated by equation 36.

The procedure just described is approximate because equation 36 does not hold

if the failure boundary is nonlinear.

A method for reducing the errors due to the nonlinearity of the failure boundary

was proposed in reference 21. Following normalization at the checking point,

relations similar in principle to equation 36 are used that correspond to the case where the failure boundary is a quadric, rather than a (hyper) plane.

The failure probability depends upon the safety index corresponding to the

space of the coordinates X? and upon the characteristics of the quadric.

The latter are determined from the condition that the quadric approximates

the nonlinear failure boundary as closely as possible at the checking point.

An alternative procedure was proposed in reference 22, in which the failure

boundary in the space of the original, non-normal variables is written in the

form

The functions Y^X-^) can be determined by approximate methods briefly described

in reference 17. Equation 44 is in effect a linearization of the failure

boundary in the space of the variables Y^. The probability distributions of

the variables Y-^ are estimated from the distributions of X^. An approximate

method for carrying out such estimates, which usually involves elementary

algebraic operations, is described in references 23 and 24. The probability distributions of Y-^ can also be estimated by Monte Carlo simulations.

Once these distributions are known, the method of normalization at the checking

point is applied to the variables Y^. The safety index, 3, is then calculated

in the space of the reduced equivalent normal variables, and equation 36 can

be applied to obtain the failure probability.

Safety Factors

Consider a structure characterized by a set of variables with means and standard

deviations X- and a Y , and checking points with coordinates X^ and Xj (i =

1 A

i 1 1

r

1, 2, ... n) in the space of the original and the reduced variables, respectively.

By definition:

n g(Xi, X2 , ..., Xn ) = E Yi(Xi) (44)

i=l

( 45 )

16
Equation 47 can be written as

X

i " YX

;L X

i

where

V 1 + V

X

i

( 46 )

(47)

The quantity Yy is termed the partial safety factor applicable to the mean of

i

the variable X^.

In design applications the means, Xj_, are seldom used, and nominal design

values, such as the 100-year wave, the allowable steel stress, Fa , or the nominal

yield stress, Fy, are employed instead. Let these nominal values be denoted by

. Equation 46 can be rewritten as [13]

x

i - Vi

where

(i — 1, 2, . < • , n) (48)

The factor Yy is the partial safety factor applicable to the nominal design

value of the variable X^.

In the particular case in which the variables of concern are the load, 0, and the

resistance, R, the partial safety factors are referred to as the load and

resistance factor. For the resistance factor the notation <?£ or p R

is used

in lieu of Y£ or y^»

From the definition of the partial safety factor (equation 47) and the definition

of the checking point in the space of the reduced variables corresponding to

Yi = &nR and Y2 = inQ, it follows that if higher order terms (see equation

27) are neglected

* exp(-aR

avR

) (50)

y~ = exp(a

0av Q

) (51)

= cos[tan_1 (V

Q

/VR

)

]

(52)

cxq = sin[ tan

- l (Vq/Vr

) ] (53)

17
where B = safety index given by equation 29.

Care must be exercised in using simplified approximate expressions for partial

safety factors. To show this, we_consider below an application of the following

expression for the load factor, Yq> proposed in reference 23 as an approximation

to equation 51 for members subjected to wind loads:

Yq = 1 + 0.55 B V

Q (54)

where B is calculated by equation 29.

Example . For members I and II of table 1, it would follow from equation

54 that Yn = 1»50 and Yn = 1«31, respectively. However, if equation

g

I _ WII _ 51 is used, Yn “ 2.32 and y n = 1.63. It is concluded that equation 4 u

I UII

may result in misleading estimates of the load factor Yq»

Equations 50 and 51 show that load and resistance factors depend not only

on the safety index, B, but on the coefficients of variation Vq, Vr as well.

For this reason, to members having the same safety index B there can correspond widely different sets of load and resistance factors as calculated

by equations 50 and 51. This is illustrated in the following example.

Example . Members I and II of table 1 have the same safety index

calculated by equation 29 (as well as approximately the same failure

probabilities — see column 10 of table 1). The values of Vr for the

two members are also the same; however the respective values of Vq differ

(table 1). It can be verified that, owing to this difference, the

load and resistance factors for the two members (given by equations 50

through 53) are <f>p - 0.88 versus <)>p - 0.83 and, as indicated earlier, K

I K

II

Yn = 2.32 versus Yn = 1*63.

^1 WII

Conversely, the use in the design of various members of the same set of load

and resistance factors does not necessarily ensure that those members will have

the same probabilities of failure. This creates difficulties in the development of

risk-consistent load and resistance factors for codified design. These difficulties,

and proposed approaches for dealing with them, are discussed in reference 13

and 15. Problems related to codes of practice are also discussed in reference

24.

18
3. SUMMARY AND CONCLUSIONS

A review was presented of fundamental topics in structural reliability that are potentially applicable to ocean engineering problems. As mentioned in the

report, although a number of studies concerned with structural systems have

been reported (e.g., references 2 to 7), the practical usefulness of such

studies remains limited, particularly as far as ocean engineering applications

are concerned. The present report is concerned with applications to individual

members

.

Some of the potential advantages of reliability methods in the context of

offshore platform analysis and design have been outlined in reference 25. In

the present report such methods have been subjected to an independent critical

review aimed at highlighting possible difficulties and pitfalls in their

application. Principal conclusions of the review are:

1. Uncertainties with respect to structural behavior and to probabilistic

characterizations of relevant parameters can render difficult, if not

impossible, meaningful comparisons between estimated safety levels of

members belonging to different types of structure or to structures subjected

to different types of load. These difficulties are compounded by the

failure of most current reliability methods to account adequately for the

complexities of systems reliability behavior, particularly in cases involving time dependent loads such as wind or waves.

2. Reliability methods based on the use of safety indices cannot be applied

in cases involving a random variable that represents a lifetime extreme

unless an explicit assumption is made with regard to the parent probability

distribution of that variable.

3. Reliability methods based on the use of safety indices or load and

resistance factors can in certain instances provide useful comparisons

between the safety levels of certain types of members. This is the case only if it can be determined that the relation between the safety index

and the failure probability for those members is independent of, or weakly

dependent upon, the relative values of the mean and coefficients of variation of the resistances and of the loads.

4. Simplified approximate expressions for partial safety factors should be

used with caution, and their range of applicability should be carefully

checked against the "exact" expressions from which they are derived.

The writers believe that the observations presented in this report can be

helpful in ensuring that reliability analyses conducted for offshore

structures can be used prudently and confidently by both practitioners and

regulatory bodies.

19
4.

REFERENCES

1. Safety Goals for Nuclear Power Plant Operation . NUREG-0880, Revision 1,

U.S. Nuclear Regulation Commission Office of Policy Evaluation, Washington,

D.C., May 1893.

2. P. Thoft - Christensen, and M. J. Baker, Structural Reliability Theory

and Applications , Springer-Verlag , Berlin, 1982.

3. F. Moses, Utilization of a Reliability-Bases API RP2A Format on Platform

Design, API PRAC Project 81-22, American Petroleum Institute, Dallas, TX,

Nov. 1982.

4. R. M. Bennett and A. H.-S. Ang, Investigation of Methods for Structural

System Reliability , UILU- ENG-83-201 4 , Dept, of Civil Engineering, University

of Illinois, Urbana, Illinois, Sept. 1983

5. C. A. Cornell, R. Rackwitz, Y. Guenard, and R. Bea, Reliabiltiy

Evaluation of Tension Leg Platforms, in Proceedings of the 4th ASCE Specialty

Conference on Probablistic Mechanics and Structural Reliability , Y.-K. Wen,

ed., January 11-13, 1984, Berkeley, Calif.

6. F. Moses, "Reliability of Structural Systems," Journal of the Structural

Division ," ASCE, Sept. 1974, pp. 1813-1820.

7. E. Vanmarcke and D. Angelides, "Risk Assessment for Offshore Structures:

A Review," Journal of Structural Engineering, Feb. 1983, pp. 555-571.

8. American National Standard Minimum Design Loads for Buildings and Other

Structures , ANSI A58. 1-1982, American National Standards Institute, New

York, N.Y., 1982.

9. T. V. Galambos, et al., "Probability Based Load Criteria: Assessment

of Current Design Practice," Journal of the Structural Division , ASCE,

May 1982, pp. 959-977.

10. R. Fleming, Wind Stresses , Engineering News, New York, 1915 (Reprints from

Engineering News , 1915).

11. Y. K. Wen, "Statistical Combination of Extreme Loads," Journal of the

Structural Division , ASCE, May 1977, pp. 1079-1095.

12. R. D. Larrabee, and C. A. Cornell, "Combination of Various Load Processes,"

Journal of the Structural Division , ASCE, January 1981, pp. 223-239.

13. B. Ellingwood et al., Development of a Probability Based Load

Criterion for American National Standard A58 , NBS Special Publication

577, National Bureau of Standards, Washington, D.C., June 1980.

20
14. A. M. Hasofer, and N. C. Lind, "Exact and Invariant Second-Moment Code

Format," Journal of the Engineering Mechanics Division, ASCE, February

1978, pp. 829-844.

15. 0. Ditlevsen, "Generalized Second Moment Reliability Index," Journal of

Structural Mechanics , Vol. 7 (1979) pp. 435-451.

16. 0. Ditlevsen, Uncertainty Modeling, McGraw-Hill International Book Company,

New York, N.Y., 1981.

17. E. Simiu, and J. R. Shaver, "Wind Loading and Reliability-Based Design,"

in Wind Engineering , Proceedings of the Fifth International Conference,

Fort Collins, Colorado, USA, July 1979, Vol. 2, Pergamon Press, OxfordNew York, 1980.

18. W. McGuire, Steel Structures , Prentice Hall, Inc., Englewood Cliffs,

N.J., 1968.

19. R. Rackwitz, and B. Fiessler, Non-normal Distributions in Structural

Reliability , SFB 96, Technical University of Munich, Ber. zur Sicherheitstheorie der Bauwerke, No. 29, 1978, pp. 1-22.

20. C. A. Cornell, "A Probability-Based Structural Code," ACI Journal , Dec. 1969.

21. B. Fiessler, H.-J Newmann, and R. Rackwitz, "Quadratic Limit States

in Structural Reliability," Journal of the Engineering Mechanics

Division , ASCE, August 1979, pp. 661-676.

22. T. V. Galambos , and M. K. Ravindra, Tentative Load and Resistance Factor

Design Criteria for Steel Buildings , Research Report No. 18, Civil

Engineering Department, Washington University, St. Louis, MO, Sept. 1973.

23. M. K. Ravindra, C. A. Cornell, and T. V. Galambos, " Wind and Snow Load

Factors for Use in LRFD," Journal of the Stuctural Division, ASCE, Sept.

1978, pp. 1443-1457.

24. F. Casciati and L. Faravelli, "Load Combination by Partial Safety Factors,"

Nuclear Engineering and Design , Vol. 75 (1982) pp. 432-452.

25. "Application of Reliability Methods in Design and Analysis of Offshore

Platforms," Journal of Structural Engineering , Oct. 1983, pp. 2265-2291.

21
.
23

figure

1.

Safety

index 0

in

space

of

variable qr.

The

cu^vt p

(qr)

represents

the

probability

density
24

Figure

2.

Failure

boundary

in

space

of

coordinates qr

andrr.
Figure 3. Failure boundary in space of coordinates u r and r r

25
>

26

Figure

4.

Failure

boundaries

for

members I

and

II
Figure 5. Probability distribution of steel yield stress

27
NBS-114A (REV. 2-80

U.S. DEPT. OF COMM. BIBLIOGRAPHIC DATA

SHEET (See instructions)

1„ PUBLICATION OR

REPORT NO.

NBSIR 84-2921

2. Performing Organ. Report No 3. Publication Date

September 1984

4. TITLE AND SUBTITLE

Structural Reliability Fundamentals and Their Application to Offshore Structures

5. AUTHOR(S)

Emil Simiu and Charles E. Smith

6. PERFORMING ORGANIZATION (If joint or other than NBS, see instructions) 7. Contract/Grant No. national bureau of standards

DEPARTMENT OF COMMERCE 8. Type of Report & Period Covered

WASHINGTON, D.C. 20234

9. SPONSORING ORGANIZATION NAME AND COMPLETE ADDRESS (Street, City . Stole, ZIP)

Minerals Management Service

Department of the Interior

Room IP-119 USGS National Ctr. Bldg., 12203 Sunrise Valley Drive

Reston, VA 22091

10. SUPPLEMENTARY NOTES

Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11.

ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bibliography or literature survey, mention it here) The objective of this report is to present an overview of fundamental topics in

structural reliability as applied to individual members, which are potentially

applicable to ocean engineering problems. These topics include: the estimation

of failure probabilities; safety indices; and safety (or load and resistance)

factors. Some of the theoretical and practical difficulties in the application

of structural reliability tools are mentioned and/or discussed.

12. KEY WORDS f Si x to twelve entries; alphabetical order; capitalize only proper names; and separate key words by serm'co/onsj

Failure; probability theory; reliability; risk; statistics; structural engineering.

13. AVAILABILITY 14. NO. OF

PRINTED PAGES

[x] Uni imited For Official Distribution. Do Not Release to NTIS

1 1 Order From Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. 31

15. Price 20402.

[X] Order From National Technical Information Service (NTIS), Springfield, VA. 22161 $8.50

U S COMM- D C 6043-P80