Reliability

The application of reliability concepts to offshore structures began as an American story. In 1979, under the direction of Dr. Fred Moses, the American Petroleum Institute began a series of studies to implement reliability design procedures for fixed offshore platforms. This would (a) offer greater uniformity in platform component reliability; (b) enable a more effective material utilisation; (c) directly account for randomness and uncertainties in engineering parameters; (d) be capable of consistent modifications to account for different location, platform type, and life; (e) offer a logical basis for incorporating new information; and (f) help to focus on research activities to emphasise areas of greatest uncertainty and have results impact reliability factors.

PRAC 79-22 Project recommended that API RP-2A standard should change its current method of checking component safety known as working stress design (WSD) format to a multiple safety factor format. This concept came to be known as the Load Resistance Factor Design or LRFD.

To illustrate this in WSD, a component is checked with an equation of the form:

$$ \frac{\sigma_n}{\gamma} > \Sigma{\sigma_i} $$

where, \( \sigma_n \) is the nominal strength, \( \gamma \) is safety factor, and \( \sigma_i \) is nominal load.

Notice only one safety factor is used in the above, and hence the reliability would depend on the range of design application. Whereas, in the LRFD approach, partial safety factors are used in the format:

$$ \phi \sigma_n > \Sigma{\gamma_i \sigma_i} $$

where, \( \phi \) is component factor, \( \Sigma{\gamma_i \sigma_i} \) is the sum of factored loads.

The research that went into producing this method included the following steps:

1. Gather statistical data on load effects and component strengths.
2. Review present performance criteria and experience.
3. Establish target reliability level for each component type, based on performance experience.
4. Calibrate load and resistance factors for tabulation in the standard.

PRAC 80-22 Project defined the reliability model for a structure as one that should incorporate safety margins and uncertainties in evaluating risk to a component or system.

Risk = 1 - Reliability = P[failure] = [Strength < Load]

The probability of component failure is as shown in Figure below.

The (extreme) load frequency curve overlapping the strength curve represents the risk. The model is viewed as a situation in which the probabilities correspond to the worst loading case — annual or lifetime, as appropriate. This overlap (i.e., probability of failure) would decrease if either (i) the mean margin of safety increases, or (ii) the uncertainty (σ) in load or resistance reduces.

The analysis of reliability is carried out by defining a failure function, g, such that g < 0 denotes failure, or:

$$ g = R - E $$

where, R is the resistance or capacity, and E is the load effect. An exact solution for the probability of failure, Pf, could be obtained if R and E are both assumed as normal distributions with respective mean values, R and E, and coefficients of variations (cov),¹ V_R, and V_E. Using these assumptions, Pf can be written as:

$$ P_f = \Phi\left( \frac{-\overline{g}}{\sigma_g} \right) $$

$$ = \Phi(-\beta) $$

where,

$$ \Phi(x) = \int_{-\infty}^{\infty} {\frac{1}{2\pi}}e^{\left({\frac{-1}{2}}z^2\right)dz} $$

However, normal distributions are not always applicable to both load and strength variables, and both R and E in turn depend on several other random variables, such as load type (gravity, wave), beam-column stability, combined hydrostatic, axial and bending strengths, etc. Therefore, a generalisation is needed to carry out realistic reliability analyses. These results are often described in terms of a safety index, β = β / σ_g, i.e., the ratio of mean safety margin to uncertainty level. In other words, β is the distance in terms of standard deviations (σ_g) from the mean (β) of the safety margin to the failure region (g < 0). When expanded, the expression becomes:

$$\beta = \frac{\ln\left({\frac{R_m}{E_m} \cdot \sqrt{\frac{1 + V_E^2}{1 + V_R^2}} } \right)}{\sqrt{\ln\left[ (1 + V_R^2) \cdot (1 + V_E^2) \right]}} $$

The LRFD method caught-on. Shell engineers extended the concept to numerous geographical areas² to develop the following table.

Table 1: Parameters of lognormal distribution of extreme 20-year load in various geographical areas (E=1.0 corresponds to E₁₀₀)

where, E₂₀ is the most probable value, and E_20m is the mean value. In addition to V_E provided above, uncertainty in hydrodynamic areas, volumes and marine growth, must be accounted for. This effect is covered by using COV = 8% and a bias factor of 1.0 (= ratio of mean value to characteristic value). This COV must be combined with COV values in table above.

Based on the GoM database, the following were derived, viz.,

$$ R_m = 1.85 $$

$$ \gamma_E = 1.35 $$

$$ cov_R = 0.05 $$

Using these above together with E_m and V_E = (0.32² + 0.08²)^0.5, the safety index, β (GOM) was computed to be 2.77, for a reference period of 20 years — based on a characteristic 100-y design event. (From β, Pf could be determined as 4E-4/y for the GoM database.)

Further, it was noted that the mean RSR is proportional to γ_E, which resulted in the expression:

$$ R_m = \frac{1.85}{1.35} \cdot \gamma_E = 1.37 \cdot \gamma_E $$

Once this was done, calibrating other locations to Gulf of Mexico safety level became easier. To extend this to a target probability of failure, first a target needed to be defined for the platform type. The industry now generally recognises it to be 3E-5/y for a manned new installation, which would be 6E-4 for 20 years (= 20 * 3E-5/y).

For the target Pf = 3E-5/y (which corresponds to P_f20 = 6E-4 for 20 years), β₂₀ could be determined as equal to 3.239. Using this safety index, probability density function plots could be generated for all tabulated regions — as I’ve done below.

Also, using this new safety index and the table above, the ISO 19902 has tabulated the mean RSR (Rm) and γ_E for the Australian Northwest Shelf and the North Sea in A.9.9.3.

So, in a gist, that’s the general theory to develop mean RSR and γ_E, which could be applied to new locations, if E_mean and V_E could be determined from Metocean and Structural response studies like Shell did in the 90s for its NS and AU assets, and later for its Southeast Asian assets. Of course, statistically extending a dataset of one region’s safety index entirely to another depends very much on how well the means and covs are computed for the new region — and in general feels both like a novel idea as well as a hack. But for our generation, this will do.

Partial action factors and reserve strength ratios

Extending the above concept to derive partial action factors and mean reserve strength ratio appropriate to the probability of failure of interest, Shell engineers documented indicative constants to represent key environments.³

Quoting from my colleagues’ article, published in Offshore Technology to describe LSM:

In 1995, Tromans and Vanderschuren introduced the load statistics method (LSM), which uses asymptotic properties of extremes to calculate most probable maxima of the governing responses of a drag dominated (fixed) structure — overturning moment and base shear. The method uses a simplified response model (i.e., a stick model), allowing the efficient and fast calculation of responses to each sea-state in a long-term hindcast database.

This enabled analysts to calculate extreme waves, loads and response-based environmental design conditions in the North Sea. The method has been used extensively for the development of response-based design conditions and reliability assessments of fixed structures in the North Sea, Gulf of Mexico and the North-West Shelf of Australia.

The LSM tool (from a suite of Metocean tools within Shell) can produce base shears and overturning moments for extreme and abnormal return periods, from which E_rp can be calculated. These formulations are from §3.7 Typical Long Term Load Distributions, EP97-5050:

$$ \alpha = \frac{\left(\frac{E_{rp}}{E_{100}} - 1\right)}{log\left(\frac{r}{100}\right)} $$

$$ A = 0.01 \cdot e^{\left(\frac{2.3026}{\alpha}\right)} $$

$$ E_0 = \frac{\alpha}{2.3026} $$

In the above,

• E_rp is the ratio of abnormal to extreme storm overturning moment (or base shear),
• E₁₀₀ is the extreme storm normalised (= 1.0), and
• r is the abnormal return period (say, 10,000 y)

This is useful especially for developing partial action factors for regions, which cannot be represented by any of the above pre-determined regions.

Table 2: Indicative constants representing key environments

Using these above, Pf could be calculated as follows:

$$ P_f = A \cdot e^{\left(\frac{-Rm}{E_0}\right)} \cdot e^{\left(\frac{(V \cdot Rm)^2}{2 \cdot E_0^2}\right)} $$

The easiest way to use the above equation is to pick a range of RSR values in terms of min. and max. (start with, say, an Rm = 1.4) and generate corresponding Pf at every small Rm interval until it exceeds 3E-5/y (33,333 years = 1 / 3E-5/y). In the plots below, the green (dashed) line indicates 2,000 years, which corresponds to Pf = 5E-4/y (L2), and the blue (dashed) line marks 33,333 years, which corresponds to Pf = 3E-5/y (L1). Here’s a summary of results.

Table 3: Mean reserve strength ratio (Rm) and corresponding partial action factor (γ_E) for probabilities of failure 3E-5/y (i.e., 33,333 year return period) and 5E-4/y (i.e., 2,000 year return period) respectively

One may notice some values in the table above that are either similar or close to those listed in §A.9.9.3.3, ISO 19902:2007.

Code to generate probability density plots

#!/usr/bin/env python3
# -*- encoding: utf-8 -*-
"""
Load and resistance probability density and safety margins as per
ISO 19902:2007. 2018 ckunte
"""
from scipy.stats import norm
import numpy as np
import matplotlib.pyplot as plt

# Function for plotting probability density by region


def pd(exp, reg, Em, Ve, Rm, Vr):
    mu_e = np.log(Em / np.sqrt(1 + Ve ** 2))  # mean
    mu_r = np.log(Rm / np.sqrt(1 + Vr ** 2))  # mean
    sigma_e = np.sqrt(np.log(1 + Ve ** 2))
    sigma_r = np.sqrt(np.log(1 + Vr ** 2))
    s_e = np.random.lognormal(mu_e, sigma_e, 1000)
    s_r = np.random.lognormal(mu_r, sigma_r, 1000)
    count_e, bins_e, ignored_e = plt.hist(
        s_e, 100, normed=True, align="mid", alpha=0.09, color="r"
    )
    count_r, bins_r, ignored_r = plt.hist(
        s_r, 100, normed=True, align="mid", alpha=0.09, color="g"
    )
    x_e = np.linspace(min(bins_e), max(bins_e), 10000)
    x_r = np.linspace(min(bins_r), max(bins_r), 10000)
    pdf_E = np.exp(
        -((np.log(x_e) - mu_e) ** 2) / (2 * sigma_e ** 2)
    ) / (x_e * sigma_e * np.sqrt(2 * np.pi))
    pdf_R = np.exp(
        -((np.log(x_r) - mu_r) ** 2) / (2 * sigma_r ** 2)
    ) / (x_r * sigma_r * np.sqrt(2 * np.pi))
    plt.plot(x_e, pdf_E, linewidth=1, color="r")
    plt.plot(x_r, pdf_R, linewidth=1, color="g")
    plt.xlim(0.25, 2.75)
    plt.ylim(0, 5.0)
    plt.title("%s - %s" % (reg, exp))
    plt.xlabel(
        "Load or resistance as times the nominal load, $x, E_{mean}(%.2f); x, R_{mean}(%.2f)$"
        % (Em, Rm)
    )
    plt.ylabel("Probability density")
    plt.axvline(x=Em, color="r", linestyle=":")
    plt.axvline(x=Rm, color="g", linestyle=":")
    plt.rcParams["grid.linestyle"] = ":"
    plt.rcParams["grid.linewidth"] = 0.5
    plt.grid(True)
    plt.savefig("pd_%s_%s.png" % (reg, exp))
    plt.close()
    pass


"""
Legend:

 exp -- Exposure level (L1 or L2)
 reg -- region
  Em -- Mean load for a reference return period
  Ve -- Load covariance
  Rm -- Mean reserve strength ratio
  Vr -- Resistance covariance

where, for CNS and SNS, NNS, GOM, AUS:

  Ve = map(sqrt(x^2 + 0.08^2), COV) => [0.2266, 0.2768, 0.3298, 0.3396]
 COV = [0.212, 0.265, 0.32, 0.33] 
  Rm = [1.73, 1.92, 1.85, 2.18] for L1 (target Pf = 3E-5/y)
  Rm = [1.40, 1.49, 1.60, 1.60] for L2 (target Pf = 5E-4/y)

"""


def main():
    # pd(exp, reg, Em, Ve, Rm, Vr)
    pd("L1", "GoM", 0.79, 0.3298, 1.85, 0.05)
    pd("L2", "GoM", 0.79, 0.3298, 1.60, 0.05)
    pd("L1", "NNS", 0.81, 0.2768, 1.92, 0.05)
    pd("L2", "NNS", 0.81, 0.2768, 1.49, 0.05)
    pd("L1", "CNS", 0.84, 0.2266, 1.73, 0.05)
    pd("L2", "CNS", 0.84, 0.2266, 1.40, 0.05)
    pd("L1", "AUS", 0.78, 0.3396, 2.18, 0.05)
    pd("L2", "AUS", 0.78, 0.3396, 1.60, 0.05)
    pass


if __name__ == "__main__":
    main()

To generate all plots in one go, just run python pdf.py.

Code to generate partial action factor and corresponding reserve strength ratio plots as per EP97-5050

#!/usr/bin/env python3
# -*- coding: UTF-8 -*-
"""Partial action factor and corresponding reserve strength ratio 
as per EP97-5050 -- 2018 ckunte

Usage: pra.py [--typ=L]
       pra.py --help
       pra.py --version

Options:
  -h, --help  Show help screen
  --typ=L     1 for partial action factor, 2 for RSR [default: 1]
  --version   Show version
"""
import numpy as np
import matplotlib.pyplot as plt
from docopt import docopt

args = docopt(
    __doc__, version="yE and corresponding Rm per EP97-5050, ver 0.2"
)
cat = int(args["--typ"])


def main():
    ## -- begin inputs --
    lbl = [
        "Australian NWS",
        "Gulf of Mexico",
        "Northern North Sea",
        "West Africa",
        "Central and Southern North Sea",
    ]
    A = [1.342, 2.13, 11.9, 19.2351, 180.0]
    E0 = [0.2041, 0.187, 0.1411, 0.1322, 0.102]
    VE = 0.07  # Cd, Cm
    VR = 0.05
    ## -- end inputs ----
    V = np.sqrt(VE ** 2 + VR ** 2)
    # RSR range (min., max.)
    x = np.linspace(1.4, 2.4)
    # Partial action factor (gamma_e)
    gamma_e = x / 1.37
    # PLot all regions
    for i, j, k in zip(A, E0, lbl):
        # Probability of failure (Pf)
        pf = i * np.exp(-x / j) * np.exp((V * x) ** 2 / (2.0 * j ** 2))
        # Return period
        rp = 1 / pf
        # Select plot type
        if cat == 1:
            # Plot Partial action factor v. Return period
            plt.plot(rp, gamma_e, linewidth=2, label=k)
            plt.ylabel("Partial action factor, $\gamma_e$")
            plt.ylim(1.02, 1.7)
        if cat == 2:
            # Plot RSR v. Return period
            plt.plot(rp, x, linewidth=2, label=k)
            plt.ylabel("Reserve strength ratio mean, $R_m$")
            plt.ylim(1.4, 2.4)
        pass
    plt.xscale("log")
    # 2,000-y ret.period (=1/1E-4)
    plt.axvline(x=2000, color="g", linestyle=":")
    # 33,333-y ret.period (=1/3E-5)
    plt.axvline(x=33333, color="b", linestyle=":")
    plt.xlabel("Return period (years)")
    plt.legend(loc=0)
    plt.savefig("pra-%d.svg" % (cat))
    pass


if __name__ == "__main__":
    main()