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Reliability-Based Strength Design of Pipelines 223

uncertainty can be reduced by additional information of the variable in terms of its statistical
significance.

Model uncertainty: This is uncertainty due to simplifications and assumptions made in
establishing the analytical model and reflects a general confidence in the applied model to
describe the real situation. It results in the difference between actual and predicted results.
Model uncertainty in a physical model for presentation of the load or resistance quantities
may be represented by a stochastic factor defined as the ratio between the true quantity and
the quantity described by the model. Guedes Soares (1997) discussed the common methods of
representing model uncertainties and illustrated principles of assessing model uncertainties.

Considering uncertainties involved in the design format, each random variable Xi can be
specified as:
Xi = B, .X, (13.4)

where XC is the characteristic value of Xi, and Bx is a normalized variable reflecting the
uncertainty in Xi.

13.3.3 Selection of Distribution Functions
Usually, the determination of the distribution function is strongly influenced by the physical
nature of the random variables. Also, its determination may be related to a well-known
description and stochastic experiment. Experience from similar problems is also very useful.
If several distributions are available, it is necessary to identify by plotting of data on
probability paper, by comparisons of moments, statistical tests, etc. Normal or lognormal
distributions are normally applied when no detailed information is available. For instance,
resistance variables are usually modeled by normal distribution, and lognormal distribution is
used for load variables. The occurrence frequency of a damage (e.g. an initial crack), is
described by Poisson distribution. Exponential distribution is used to model the capacity of
detecting a certain damage.

13.3.4 Determination of Statistical Values
Statistical values used to describe a random variable are mean value and coefficient of
variation (COV). These statistical values shall normally be obtained from recognized data
sources. Regression analysis may be applied based on methods of moment, least-square fit
methods, maximum likelihood estimation technique, etc.

13.4 Calibration of Safety Factors

13-4.1 General

One of the important applications of structural reliability methods is to calibrate safety factors
in design format in order to achieve a consistent safety level. The safety factors are
determined so that the calibrated failure probability, Pf,i for various conditions is as close to
the target reliability level Pl as possible: