Comparative reliability analysis of ships under vector-load processes

Abstract

This paper deals with reliability analysis of ships under vector-load processes. Failure is expressed by multiple response variables. Different methods can be used to assess failure probability, primary among them are the time-independent and out-crossing (up-crossing) rate methods. This paper describes the procedure for performing reliability analysis of ships under vector-load processes by the out-crossing rate method. Alternative methods based on piecewise linear or continuous modeling of failure surface are explored for calculating the out-crossing rate and compared. The methods are exemplified by calculating the conditional probability of a damaged double hull oil tanker under combined vertical and horizontal bending moments.

Appendices

Appendix I: Up-crossing rate by Naess’s method for a problem with limit state defined by Eq. 29

According to Eqs. 5 and 29,

$$ H\left( {{\mathbf{Q}}(t)} \right) = \left( {{\frac{{M_{v} (t)}}{{M_{uv} }}}} \right)^{\alpha } + \left( {{\frac{{M_{h} (t)}}{{M_{uh} }}}} \right)^{\beta } = \left( {X(t)} \right)^{\alpha } + \left( {Y(t)} \right)^{\beta } \quad 0 \le X,Y \le 1 $$

(32)

where ${\mathbf{Q}}(t) = \left\{ {X(t),Y(t)} \right\}^{T} = \left\{ {{\frac{{M_{v} (t)}}{{M_{uv} }}},{\frac{{M_{h} (t)}}{{M_{uh} }}}} \right\}^{T}$

From Eq. 32, we can derive that

$$ \dot{H} = \alpha X^{\alpha - 1} \dot{X} + \beta Y^{\beta - 1} \dot{Y} $$

(33)

Write the mean vector of Q as μ _Q and the covariance matrix Σ of $\left( {{\mathbf{Q}}^{T} ,{\dot{\mathbf{Q}}}^{T} } \right)^{T}$ as

$$ \Upsigma = \left( {\begin{array}{*{20}c} {\Upsigma_{11} } & {\Upsigma_{12} } \\ {\Upsigma_{21} } & {\Upsigma_{22} } \\ \end{array} } \right) $$

(34)

Given ${\mathbf{Q}} = {\mathbf{q}} = (x,y)^{T}$, the joint PDF of $\dot{Q}$, is multivariate normal with mean value ${\varvec{\upmu}}_{c} = {\varvec{\Upsigma}}_{21} {\varvec{\Upsigma}}_{11}^{ - 1} ({\mathbf{q}} - {\varvec{\upmu}}_{Q} ) = (\mu_{1} ,\mu_{2} )^{T}$ and covariance matrix ${\varvec{\Upxi}} = {\varvec{\Upsigma}}_{22} - {\varvec{\Upsigma}}_{21} {\varvec{\Upsigma}}_{11}^{ - 1} {\varvec{\Upsigma}}_{12} .$ Hence the statistics of $\{ \dot{H}|{\mathbf{Q}} = {\mathbf{q}}\}$ are given as:

$$ \mu = \alpha x^{\alpha - 1} \mu_{1} + \beta y^{\beta - 1} \mu_{2} = {\mathbf{p}}^{T} {\varvec{\upmu}}_{c} $$

(35)

$$ \sigma^{2} = \alpha^{2} x^{2(\alpha - 1)} \Upxi_{11} + \beta^{2} y^{2(\beta - 1)} \Upxi_{22} + \alpha \beta x^{\alpha - 1} y^{\beta - 1} \left( {\Upxi_{12} + \Upxi_{21} } \right) $$

(36)

where ${\mathbf{p}}^{T} = \left[ {\alpha x^{\alpha - 1} ,\beta y^{\beta - 1} } \right]$

Furthermore,

$$ x(y,h) = (h - y^{\beta } )^{1/\alpha } $$

(37)

$$ \left| {{\frac{\partial x}{\partial h}}} \right| = {\frac{1}{\alpha }}(h - y^{\beta } )^{{{\frac{1}{\alpha }} - 1}} = {\frac{1}{\alpha }}\left[ {x(y,h)} \right]^{1 - \alpha } $$

(38)

Equation 20 becomes:

$$ v\left( \xi \right) = \int\limits_{{y_{l} (\xi )}}^{{y_{u} (\xi )}} {{\frac{1}{\alpha }}\tilde{x}^{1 - \alpha } \cdot \left[ {\tilde{\sigma }\phi \left( {{\frac{{\tilde{\mu }}}{{\tilde{\sigma }}}}} \right) + \tilde{\mu }\Upphi \left( {{\frac{{\tilde{\mu }}}{{\tilde{\sigma }}}}} \right)} \right] \cdot {\frac{1}{{2\pi \sqrt {\left| {{\varvec{\Upsigma}}_{11} } \right|} }}}\exp \left( { - {\frac{1}{2}}({\tilde{\mathbf{q}}} - {\varvec{\upmu}}_{Q} )^{T} {\varvec{\Upsigma}}_{11}^{ - 1} ({\tilde{\mathbf{q}}} - {\varvec{\upmu}}_{Q} )} \right){\text{d}}y} $$

(39)

where ~ indicates that $x = x(y,\xi ).$ Set

$f_{1} = {\frac{1}{\alpha }}\tilde{x}^{1 - \alpha } ,\;f_{2} = \left[ {\tilde{\sigma }\phi \left( {{\frac{{\tilde{\mu }}}{{\tilde{\sigma }}}}} \right) + \tilde{\mu }\Upphi \left( {{\frac{{\tilde{\mu }}}{{\tilde{\sigma }}}}} \right)} \right],f_{3} = {\frac{1}{{2\pi \sqrt {\left| {{\varvec{\Upsigma}}_{11} } \right|} }}}\exp \left( { - {\frac{1}{2}}({\tilde{\mathbf{q}}} - {\varvec{\upmu}}_{Q} )^{T} {\varvec{\Upsigma}}_{11}^{ - 1} ({\tilde{\mathbf{q}}} - {\varvec{\upmu}}_{Q} )} \right),$ and f _t  = f₁ · f₂ · f₃, 

$$ v\left( \xi \right) = \int\limits_{{y_{l} (\xi )}}^{{y_{u} (\xi )}} {f_{t} {\text{d}}y} = \int\limits_{{y_{l} (\xi )}}^{{y_{u} (\xi )}} {f_{1} \cdot f_{2} \cdot f_{3} {\text{d}}y} $$

(40)

For the limit surface given in Eq. 29, the out-crossing rate is given by Eq. 39 with level ξ = 1. The integration in Eq. 39 could be evaluated numerically. However, an asymptotic evaluation for high level mean up-crossing can also be achieved by the generalized Laplace method.

Appendix II: asymptotic evaluation of high level mean up-crossing rate defined by Eq. 39

To calculate the asymptotic mean up-crossing rate by the Laplace method, Eq. 39 should be simplified first. The following variables are defined, $c(y) = {\frac{1}{2}}\left( {{\tilde{\mathbf{q}}} - {\varvec{\upmu}}_{Q} } \right)^{T} {\varvec{\Upsigma}}_{11}^{ - 1} \left( {{\tilde{\mathbf{q}}} - {\varvec{\upmu}}_{Q} } \right),a(y) = \alpha \tilde{x}^{\alpha - 1}$ and $b(y) = {\frac{{\tilde{\sigma }}}{{\text{cov} (\dot{X},Y)a(y)}}},$ then

$$ c' = {\frac{{{\text{d}}c(y)}}{{{\text{d}}y}}} = \left( {{\frac{{{\text{d}}{\tilde{\mathbf{q}}}}}{{{\text{d}}y}}}} \right)^{T} {\varvec{\Upsigma}}_{11}^{ - 1} ({\tilde{\mathbf{q}}} - {\varvec{\upmu}}_{Q} ) = \left[ { - {\frac{{\beta y^{\beta - 1} }}{{\alpha \tilde{x}^{\alpha - 1} }}},1} \right]{\varvec{\Upsigma}}_{11}^{ - 1} ({\tilde{\mathbf{q}}} - {\varvec{\upmu}}_{Q} ) $$

(41)

$$ \tilde{\mu } = {\tilde{\mathbf{p}}}^{T} {\varvec{\Upsigma}}_{21} {\varvec{\Upsigma}}_{11}^{ - 1} ({\tilde{\mathbf{q}}} - {\varvec{\upmu}}_{Q} ) = \left[ {\alpha \tilde{x}^{\alpha - 1} ,\beta y^{\beta - 1} } \right]\text{cov} (\dot{X},Y)\left( {\begin{array}{*{20}c} 0 & 1 \\ { - 1} & 0 \\ \end{array} } \right){\varvec{\Upsigma}}_{11}^{ - 1} ({\tilde{\mathbf{q}}} - {\varvec{\upmu}}_{Q} ) = \text{cov} (\dot{X},Y)a(y)c' $$

(42)

$$ v = {\frac{{\text{cov} (\dot{X},Y)}}{{2\pi \sqrt {\left| {{\varvec{\Upsigma}}_{11} } \right|} }}}\int\limits_{{y_{l} (\xi )}}^{{y_{u} (\xi )}} {\left[ {b\phi \left( {{\frac{c'}{b}}} \right) + c'\Upphi \left( {{\frac{c'}{b}}} \right)} \right] \cdot \exp ( - c){\text{d}}y} $$

(43)

Integration by parts on the second term of Eq. 43 gives:

$$ \begin{aligned} v & = {\frac{{\text{cov} \left( {\dot{X},Y} \right)}}{{2\pi \sqrt {\left| {{\varvec{\Upsigma}}_{11} } \right|} }}}\left( {\int_{{y_{l} (\xi )}}^{{y_{u} (\xi )}} {\left[ {b + \left( {{\frac{c'}{b}}} \right)^{'} } \right]\phi \left( {{\frac{c'}{b}}} \right)\exp ( - c){\text{d}}y} - \left[ {\Upphi \left( {{\frac{c'}{b}}} \right) \cdot \exp ( - c)} \right]_{{y_{l} (\xi )}}^{{y_{u} (\xi )}} } \right) \\ & = {\frac{{\text{cov} \left( {\dot{X},Y} \right)}}{{2\pi \sqrt {\left| {{\varvec{\Upsigma}}_{11} } \right|} }}}\left( {{\frac{1}{{\sqrt {2\pi } }}}\int_{{y_{l} (\xi )}}^{{y_{u} (\xi )}} {\left( {b + {\frac{c''}{b}} - {\frac{c'b'}{{b^{2} }}}} \right)\cdot\exp \left( { - \left( {c + {\frac{1}{2}}\left( {{\frac{c'}{b}}} \right)^{2} } \right)} \right){\text{d}}y} - \left[ {\Upphi \left( {{\frac{c'}{b}}} \right) \cdot \exp ( - c)} \right]_{{y_{l} (\xi )}}^{{y_{u} (\xi )}} } \right) \\ \end{aligned} $$

(44)

Set $f(y,\xi ) = b + {\frac{c''}{b}} - {\frac{c'b'}{{b^{2} }}}$ and $k(y,\xi ) = c + {\frac{1}{2}}\left( {{\frac{c'}{b}}} \right)^{2}$. The integration in Eq. 44 can be calculated by the Laplace method which gives the asymptotic evaluation of the integration involving a large quantity ξ as following:

$$ \int\limits_{{y_{l} }}^{{y_{u} }} {f(y,\xi )\exp ( - k(y,\xi )){\text{d}}y} = \left\{ {\begin{array}{*{20}c} {\sqrt {2\pi } f(y_{0} ,\xi )\exp ( - k(y_{0} ,\xi ))/\sqrt {k''(y_{0} ,\xi )} \quad y_{l} \le y_{0} \le y_{u} ,k'(y_{0} ,\xi ) = 0} \\ {f(y_{l} ,\xi )\exp ( - k(y_{l} ,\xi ))/k'(y_{l} ,\xi )\quad y_{0} = y_{l} ,k'(y_{0} ,\xi ) \ne 0} \\ { - f(y_{u} ,\xi )\exp ( - k(y_{u} ,\xi ))/k'(y_{u} ,\xi )\quad y_{0} = y_{u} ,k'(y_{0} ,\xi ) \ne 0} \\ \end{array} } \right. $$

(45)

where y ₀ is found numerically to give the global minimum of k(y, ξ) within the integration interval [y _l , y _u ] and f(y ₀, ξ) ≠ 0. Notice that

$$ k'(y,\xi ) = {\frac{c'(y,\xi )}{b(y,\xi )}}f(y,\xi ) $$

(46)

If c(y, ξ) and k(y, ξ) have the same global minimum, the right hand side of Eq. 45 becomes:

$$ \left\{ {\begin{array}{*{20}l} {\left( {b(y_{0} ,\xi ) + {\frac{{c''(y_{0} ,\xi )}}{{b(y_{0} ,\xi )}}}} \right)\exp \left( { - c(y_{0} ,\xi )} \right)/\sqrt {c''(y_{0} ,\xi ) + \left( {c''(y_{0} ,\xi )/b(y_{0} ,\xi )} \right)^{2} } \quad y_{l} \le y_{0} \le y_{u} ,c'(y_{0} ,\xi ) = 0} \\ {{\frac{1}{{\sqrt {2\pi } }}}\;b\left( {y_{l} ,\xi } \right)\exp ( - k(y_{l} ,\xi ))/c'\left( {y_{l} ,\xi } \right)\quad y_{0} = y_{l} ,c'\left( {y_{0} ,\xi } \right) \ne 0} \\ { - {\frac{1}{{\sqrt {2\pi } }}}\;b\left( {y_{u} ,\xi } \right)\exp \left( { - k(y_{u} ,\xi )} \right)/c'\left( {y_{u} ,\xi } \right)\quad y_{0} = y_{u} ,c'(y_{0} ,\xi ) \ne 0} \\ \end{array} } \right. $$

(47)

Then the asymptotic mean up-crossing rate can be calculated using Eqs. 44 and 47.