5.1 Uncertainty Associated with the Design Variables.
As we have seen from the material in the preceding sec-tions, there is a certain degree of randomness or uncer-tainty in our ability to predict both the loads imposed on the ship’s structure (the demand) and the ability of the structure to withstand those loads (the capability).
The sources of these uncertainties include phenomena that can be measured and quantified but cannot be per-fectly controlled or predicted by the designer, and phe-nomena for which adequate knowledge is lacking. The term objective uncertainties is sometimes applied in scribing the former, and the term subjective is used in de-scribing the latter. In principle, the objective uncertain-ties can be expressed in statistical terms, using available data and theoretical procedures. The subjective uncer-tainties, which are known to exist but which cannot be fully quantified as a result of a lack of knowledge, must be dealt with through judgment and the application of factors of safety.
An example of an objective uncertainty is the variabil-ity in the strength properties of the steel used in con-structing the ship. The magnitude of this variability is controlled to some extent through the practices of spec-ifying minimum properties for the steel, and then test-ing the material as produced by the steel mill to ensure compliance with the specifications. Departures from the specified properties may exist for several reasons. For practical reasons, the sampling and testing cannot be ap-plied to all of the material going into the ship, but only to a limited sampling of the material. As a result of slight variations in its manufacturing experience, some of the material may exhibit different properties from those of material manufactured by supposedly identical proce-dures. After arrival in the shipyard, the material prop-erties may be altered by the operations such as cutting, forming, and welding, which are involved in building it into the ship. These variations in properties may be re-duced by a more rigorous system of testing and quality control, all of which adds to the final cost of the ship.
A compromise must therefore be reached between cost and the level of variation or uncertainty that is consid-ered acceptable and that may be accommodated by the degree of conservatism in the design.
On the other hand, the subjective uncertainty cannot be quantified on the basis of direct observation or analyt-ical reasoning but must be deduced by indirect means.
The most common source of this uncertainty is a defi-ciency in the understanding of a fundamental physical phenomenon or incomplete development of the mathe-matical procedures needed for the purpose of predict-ing a certain aspect of the structural response. An ex-ample of incomplete theoretical knowledge is the small amplitude limitation inherent in the linear theory of wave loads and ship motions, as outlined in Section 2.6 of this
book. An analogous limitation exists in the application of linear elastic theory to the prediction of the structural response. Even though there have been important ad-vances in theoretical and computational methods of non-linear structural analysis, there is still a significant ele-ment of uncertainty in predictions of structural behavior in the vicinity of structural collapse. In this region, non-linear material behavior as well as nonnon-linear geometric effects are present, and the overall response may involve the sequential interaction of several elementary response phenomena. Of course, it is the goal of ongoing research to change these subjective uncertainties into objective uncertainties.
Therefore, it is clear that the design of ship structures must take into account the uncertainties in the predic-tions of both demand and capability of strength. To ar-rive at the most efficient structure that will achieve an acceptable degree of reliability, it is necessary to attempt to quantify the uncertainties and allow for their possible magnitudes and consequences. Reliability assessment of-fers an excellent means for doing this.
A typical reliability assessment procedure for marine structures is described in Fig. 133. Starting with a config-uration of the marine structure and using random ocean waves as input, the wave loads acting on the structure can be determined using spectral analysis, as discussed in Section 2.6 (refer to Fig. 133). Generally, for design analysis the most important loads are the large ones. Ex-trapolation procedures are usually used to determine the characteristics of these large loads. For example, in the case of ocean-going vessels this is done either through the determination of a long-term distribution of the wave loads or through the evaluation of an extreme load distri-bution that may occur in specified storm conditions (see Sections 2.6 and 2.7).
In general, wave loads acting on an ocean-going ves-sel include low-frequency loads due to the motion of the vessel in waves as a rigid body. They also include higher-frequency loads (and response) due to slamming and springing, which can be determined by consider-ing the ship as a flexible body. In principle, these loads should be combined stochastically to determine the total wave load, as discussed in Section 2.6. Referring back to Fig. 133, other loads besides wave loads occur on a ma-rine structure. These loads may be important in magni-tude, though usually less random in nature (except possi-bly for wind loads on offshore structures). For example, in the case of ocean-going vessels, these loads consist mainly of still water loads and thermal loads.
Following Fig. 133, the response of the marine struc-ture to the total combined loads is determined and com-pared with the resistance or capability of the structure.
This comparison may be conducted through one of sev-eral reliability methods, which will be discussed later.
Configuration of the Structure
Random
Ocean Waves Wave Loads on
the Structure
Structural Capability or
Strength
Structural Response to Combined Extreme Loads
Reliability Analysis -Safety Indices & Probability
of Failure
Comparison with Acceptable Safety Indices
& Probability of Failure
End yes
Acceptable? no
Start New Cycle Other Loads on
the Structure
Combined Extreme Loads Extreme Wave
Loads
Fig. 133 Probabilistic analysis of marine structures.
Based on these methods, safety indices or probabilities of failure are estimated and compared with acceptable values. A new cycle may be necessary if the estimated indices are below the acceptable limits.
5.2 Basic Reliability Concept. To illustrate some as-pects of the procedure described in Fig. 133 and to introduce the basic concept in the reliability analy-sis, the following example is given. Consider a sim-ple beam subjected to a loading induced by the envi-ronment (e.g., wave load). Traditionally, in the design of such a beam designers have used fixed determinis-tic values for the load acting on the beam and for its strength. In reality, these values are not unique values but rather have probability distributions that reflect un-certainties in the load and the strength of the beam.
Structural reliability theory deals mainly with the assess-ment of these uncertainties and the methods of quantify-ing and rationally includquantify-ing them in the design process.
The load and the strength are thus modeled as random variables.
Figure 134 shows the probability density functions (PDF’s) of the load and the strength of the beam in terms of applied bending moment and ultimate moment capac-ity of the beam, respectively. Both the load, Z, and the strength, S, are assumed in this example to follow the
fz(z)
fs(s)
Area = Fs(z)
z = z µs s or z µz
Fig. 134 Load and strength probability density functions.
normal (Gaussian) probability distribution with mean values of µZ = 60.74 MN-m (20,000 ft-ton) and µS = 91.11 MN-m (30,000 ft-ton), respectively, and standard deviations of σZ = 7.59 MN-m (2,500 ft-ton) and σS = 9.11 MN-m (3,000 ft-ton), respectively.
A simple function, g(s,z), called the limit state func-tion, can be constructed that describes the safety margin, M, between the strength of the beam and the load acting on it:
M = g(s, z) = S − Z (272) Both S and Z are random variables and may assume sev-eral values. Therefore, the following events or conditions describe the possible states of the beam:
(i) M = g(s,z) < 0 represents a failure state because this means that the load Z exceeds the strength S.
(ii) M = g(s,z) > 0 represents a safe state.
(iii) M= g(s,z) = 0 represents the limit state surface (line, in this case) or the border line between the safe and failure states.
The probability of failure implied in state (i) can be com-puted from integration is over all values of s and z where the margin Mis not positive (i.e., not in the safe state). If the applied load on the beam is statistically independent from the beam strength, the equation (273) can be simplified and interpreted easily as re-spectively, both of which in this example are Gaussian.
Equation (274) is the convolution integral with respect to z and can be interpreted with reference to Fig. 93. If Z= z (i.e., the random variable Z is equal to a specific value z), the conditional probability of failure would be FS(Z). But because z< Z ≤ z + dz is associated with probability fZ(z)dz, integration of all values of z results in equation (274).
In our example, S and Z are both statistically inde-pendent and normally distributed. Equation (274) can be thus shown to reduce to
pf = (−β) (275)
where is the standard normal cumulative distribution function (CDF) andβ is called a safety index, defined as
β = µs− µz
"
σs2+ σz2
(276)
—see plot of equation (275) in Fig. 135. Notice that as the safety index β increases, the probability of failure
pf as given by equation (275) decreases. The safety of the beam as measured by the safety indexβ can be thus increased—see equation (276)—by increasing the differ-ence between the meansµS− µZor decreasing the stan-dard deviations σS and σZ. Substituting into equation (276) the numerical values forµS,µZ,σS, andσZ given in our simple beam example results in a safety index β = 2.56. Equation (275) can be then used in conjunction with tables of standard normal CDF to yield a probability of failure= 5.23 × 10−3.
Typical reliability analysis of complex structures is more complicated than the simple example given here, for several reasons. The probability distributions of some or all the variables may not be normal, therefore equa-tion (275) would be no longer valid. However, a trans-formation to normal variables is possible with various degrees of approximation. The limit state equations may contain more than two variables and may not have a lin-ear form as given by equation (272). Finally, a complex structure will generally consist of several members and each member may fail in one of several modes of failure, thus requiring system reliability instead of the “member”
reliability described in the previous example. A detailed discussion of these aspects can be found in Mansour (1989).
The reliability concept can be extended to several ran-dom variables, instead of only two, as given by equation (274). For the case of n random variables, the probabil-ity of failure (or generally, the probabilprobabil-ity of exceeding a specified limit state) can be determined from
pf =
· · ·
fx(x1, x2. . . xn)dx1dx2. . . dxn (277) where fx is the JPDF of the design random variables x1. . . xn. The domain of the multiple integration in equa-tion (277) is over the unsafe region of a limit state associated with the structure. The limit state function g(x1. . . xn) is formulated such that the unsafe region, the integration domain of equation (277), is given by
g(x1. . . xn)≤ 0
for which the limit state equation reduces to equation (272) for the simple two-variable case. From the com-puted probability of failure, the generalized safety index is defined by:
β = −1(1− pf) (278) This level of reliability assessment (usually titled Level 3, or direct integration method) is limited in terms of applications to complex marine structures. However, it can still be applied to assess ship hull girder primary strength because in this case, the ship is usually con-sidered as a beam. Examples of such applications can be found in Mansour (1972) and Mansour and Faulkner (1972).
The preceding discussion indicates that certain spe-cific load and strength information are necessary for performing a reliability analysis of marine structures. It
10-9 10-8 10-7 10-6 10-5 10-4 10-3 10-2 10-1
Standard Normal Variate,z
2 3 4 5 6
Safety Index, β
Probablility of Failure, Pf
− fz
z
Fig. 135 Safety index versus probability of failure.
is mostly in this area that reliability analysis of marine structures differs from typical civil engineering struc-tures. Prior to estimating the loads acting on ships or marine structures, a statistical representation of the en-vironment is necessary. This includes waves, wind, ice, seismic activity, and current. (The last four items are more important for fixed offshore structures than for floating vessels.) The environmental information can then be used as input to determine the loads acting on the structure. Typically, an input/output spectral analy-sis procedure is used to determine the “short-term” loads in a specific sea condition (stationary condition)—see Section 2.6. The required transfer function is determined from strip theory using the equations of motion of the vessel or from a towing tank experiment. In offshore structures, Morrison’s equation is usually used to deter-mine the wave-load transfer function.
Short-term prediction of the loads is not sufficient for the reliability analysis. Extreme values and long-term prediction of the maximum loads and their statistics are more valuable. For this purpose, order statistics and
statistics of extremes play a very important role. Gum-bel’s theory of asymptotic distributions is often used in this regard. In the long-term prediction, the fatigue loads (i.e., the cyclic repetitive loads that cause cumulative damage to the structure) must also be considered. For a complete description of this aspect of reliability analysis, methods of combining the loads such as static and dy-namic, including high- and low-frequency loads, must be considered. In nature, many of these loads act simultane-ously, therefore their combination must be evaluated for a meaningful reliability analysis (see Section 2.6).
The second major component in the reliability analysis is the strength (or resistance) of the marine structure and the evaluation of its modes of failure. In this regard, sev-eral limit states may be defined such as the ultimate limit state, fatigue limit state, and serviceability limit state.
The first relates to the maximum load-carrying capacity of the structure, the second to the damaging effect of re-peated loading, and the third to criteria governing nor-mal use and durability. Each of these limit states may include several modes—for example, the ultimate limit
state includes excessive yielding (plastic mechanisms) and instability (buckling failure).
5.3 Approximate Methods of Reliability Analysis. As discussed in the previous section, the direct integration method of reliability analysis (Level 3) can be difficult to apply in practice. The two main reasons for this are the lack of information to determine the joint probability density function (JPDF) of the design variables and the difficulty associated with the evaluation of the resulting multiple integral. For these reasons, approximate meth-ods of reliability analysis were developed, most of these are classified as “Level 2” reliability analysis. These meth-ods are:
r
Mean value first order second moment method (MV-FOSM)r
First order reliability methods (FORM)r
Second order reliability methods (SORM)r
Advanced mean value method (AMV)r
Adaptive importance sampling (AIS)r
Monte Carlo simulation (MCS)Some of these methods are discussed following.
5.3.1 Mean Value First Order Second Moment (MV-FOSM). If Z is a random variable representing the load and S is a random variable representing the strength, then the safety margin as defined previously is
M= S − Z (279)
Failure occurs when the total applied load Z exceeds the ultimate capacity S, that is, when the margin M is nega-tive. Therefore, the probability of failure (Ang and Tang, 1975), pf, is
pf = P[M ≤ 0] = FM(0) (280) where FM is the cumulative distribution function of the margin M.
For statistically independent load Z and strength S, the mean µmand variance σ2mof the margin are given by
µm= µs− µz
σm2 = σs2+ σz2 (281) The standardized margin G, which has a zero mean and a unit standard deviation, can be written as
G= M− µm
σm
(282) Failure occurs (or a limit state is exceeded) when M ≤ 0 so that equation (282) can be written as
pf = FM(0)= FG
−µm
σm
= FG(−β) (283) whereβ is the safety index, which is the inverse of the coefficient of variation of the safety margin. Thus in the MVFOSM method, a safety indexβ is defined as
β = µm
σm
(284)
If the distribution function FG is known, then the ex-act probability of failure associated with the safety index can be determined. But even for unknown or unspecified distribution function FG, there will be a corresponding though unspecified probability of failure for each value ofβ. Thus, β may be taken as a safety measure, as is the case in the MVFOSM method.
The foregoing results can be generalized as follows.
Define a limit state (or performance) function g as M = g(x1, x2. . . xn) (285) where xi are the load and strength parameters consid-ered as random variables, and the limit state function g is a function that relates these variables for the limit state of interest (serviceability or ultimate state). The limit state is exceeded (failure) when
M = g(x1, x2. . . xn)≤ 0 (286) Notice that equation (286) is the same as the integra-tion domain in the Level 3 reliability—see equaintegra-tion (277).
The limit state function can be expanded using Taylor’s series, and if only the first-order terms are retained, we get where xi∗is the linearization point and the partial deriva-tives are evaluated at that point, see Ang and Tang, 1975.
In the MVFOSM method, the linearization point is set at the mean values ( ¯x1, ¯x2, . . . ¯xn). The mean and variance of Mare then approximated by
µm∼= g( ¯x1, ¯x2, . . . ¯xn) (288)
where ρxixj is the correlation coefficient and the sub-scripts xiand xj denote evaluation of the partial deriva-tives at the mean point. The accuracy of equations (288) and (289) depends on the effect of neglecting the higher-order terms in equation (287).
If the variables xi are statistically uncorrelated, then equation (288) remains unchanged, but equation (289) becomes
As an example, if the margin M is represented by the variables S and Z only, that is,
M = g(x1,x2)= g(S,Z) = S − Z
then applying equations (288) and (290) for determining the mean and variance, one immediately obtains iden-tical results as given by equation (281). This method is called the MVFOSM method because the linearization of
Failure Region
Safe Region M=0
Z'
S' D
0
Fig. 136 Limit state function in the space of reduced variates.
the limit state function takes place at the mean value (MV); only the first-order (FO) terms are retained in Tay-lor’s series expansion, and up to the second moment (SM) of the random variables (means and variances) are used in the reliability measure rather than their full prob-ability distributions.
A geometric interpretation (see Ang and Tang, 1975) of the safety margin M= S − Z will be particularly use-ful for the discussion of the first order reliability method (FORM), which will be presented later. First, we notice that M > 0 represents a safe state or region, M < 0 rep-resents a failure state, and M= 0 represents a limit state or failure surface (or line, in the case of two variables).
The standard or “reduced” variates of S and Z can be written as
S = S− µs
σs
; Z = Z− µz
σz
Therefore, the limit state function, M= 0, can be written in the space of reduced variates as
M = σsS − σzZ + µs− µz= 0 which is a straight line, shown in Fig. 136.
The region on one side of the straight line that contains the origin represents the safe state (M> 0), and the other region represents the failure state (M< 0). Thus, the dis-tance from the origin to the line M = 0 can be used as a measure of reliability. In fact, from geometry the mini-mum distance, D, shown in Fig. 136, is given by (Ang and Tang, 1975):
D= µs− µz
"
σs2+ σz2
Notice that D is equal to the safety indexβ for the case of the normal variates and linear limit state function dis-cussed earlier, that is, for this case,
Notice that D is equal to the safety indexβ for the case of the normal variates and linear limit state function dis-cussed earlier, that is, for this case,