Dynamic reliability under random shocks

Abstract

The theory of dynamic reliability is extended to explicitly incorporate random changes of the state variables at the time points of transition between the discrete states of the Markovian component of the model. It is shown, that this can be used to model random loads induced by external events, a situation often encountered in structural reliability problems. A state model is developed, which represents the behavior of elementary structures degrading, due to aging processes and stochastic environmental loads. Finally, the structural reliability of the basic components of underground water pipe networks is estimated using the proposed framework.

Introduction

Traditionally, state graph processes are formulated in terms of state probabilities. This means, a system of equations is derived, which can be used to calculate state probabilities. Solving this system means solving the process. This also holds for dynamic reliability, where the state probabilities $\text{p}{}_{\mathrm{j}}\text{(}\text{x}\text{,t)}$ are given by

$$\text{p}{}_{\mathrm{j}}\text{(}\text{x}\text{,t)=}\text{∫}\text{(}\text{u}\text{)}\text{p}{}_{\mathrm{j}}\text{(}\text{u}\text{,0)δ(}\text{x}\text{−}\text{g}{}_{\mathrm{j}}\text{(t,}\text{u}\text{))}\text{exp}\text{−}\text{∫}\text{0}\text{t}\text{λ}{}_{\mathrm{j}}\text{(}\text{g}{}_{\mathrm{j}}\text{(s,}\text{u}\text{))}\text{d}\text{s}\text{d}\text{u}\text{+}\text{∑}\text{(i)}\text{∫}\text{0}\text{t}\text{∫}\text{(}\text{u}\text{)}\text{δ(}\text{x}\text{−}\text{g}{}_{\mathrm{j}}\text{(t−τ,}\text{u}\text{))λ}{}_{\mathrm{ij}}\text{(}\text{u}\text{)}\text{exp}\text{−}\text{∫}\text{0}\text{t−τ}\text{λ}{}_{\mathrm{j}}\text{(}\text{g}{}_{\mathrm{j}}\text{(s,}\text{u}\text{))}\text{d}\text{s}\text{p}{}_{\mathrm{i}}\text{(}\text{u}\text{,τ)}\text{d}\text{u}\text{dτ}$$

where x is a vector of physical state variables, which describe the physical state of the system. The evolution of x over time can be expressed for a given state and initial conditions u by the vector function $\text{g}{}_{\mathrm{j}}\text{(t,}\text{u}\text{).}$ Markovian transitions between the elements of a discrete state space are described by transition rates λ_ij, which in turn may depend on x. The sum of all transition rates leaving a certain state j is called λ_j. This equation is derived in Ref. [7] and further analyzed in [8], [9], [10]. A more practical application is given in Ref. [17].

A system fails when either its physical variables enter a failure domain or the system reaches a failure state [15]. Let S_jF be the failure region of state j and $\text{λ}{}_{\mathrm{j}}\text{(}\text{x}\text{)=0}$ for $\text{x}\text{∈S}{}_{\mathrm{<mtext>j</mtext><mtext>F</mtext>}}\text{.}$ If we assume the dynamics vanish in S_jF then failure probability $\text{R}\text{̄}\text{(t)}$ is given by:

$$\text{R}\text{̄}\text{(t)=}\text{∫}\text{0}\text{t}\text{∫}\text{(}\text{x}\text{)}\text{∑}\text{j∈X}\text{p}{}_{\mathrm{j}}\text{(}\text{x}\text{,t′)}\text{∑}\text{i∈Y}\text{λ}{}_{\mathrm{ji}}\text{(}\text{x}\text{)+δ(t′−t)H}{}_{\mathrm{<mtext>j</mtext><mtext>F</mtext>}}\text{(}\text{x}\text{)}\text{d}\text{x}\text{d}\text{t′}$$

where X is the set of working states and Y the set of failed states. $\text{H}{}_{\mathrm{<mtext>j</mtext><mtext>F</mtext>}}\text{(}\text{x}\text{)}$ is the indicator function of failure domain, defined by:

$$\text{H}{}_{\mathrm{<mtext>j</mtext><mtext>F</mtext>}}\text{(}\text{x}\text{)=}\text{1}\text{if}\text{x}\text{∈S}{}_{\mathrm{<mtext>j</mtext><mtext>F</mtext>}}\text{0}\text{else}$$

The second way leading to a failure, i.e. an entering into the failure region S_jF, has no correspondent in classical reliability. It has to be noticed that on the contrary, crossing the border of the failure region is usually the only way of failure in structural reliability problems. Eliminating failed states simplifies Eq. (2) to:

$$\text{R}\text{̄}\text{(t)=}\text{∫}\text{(}\text{x}\text{)}\text{∑}\text{j=1}\text{N}\text{p}{}_{\mathrm{j}}\text{(}\text{x}\text{,t)H}{}_{\mathrm{<mtext>j</mtext><mtext>F</mtext>}}\text{(}\text{x}\text{)}\text{d}\text{x}$$

where N is the number of states.

What happens, if a dynamic system experiences external loads? Although the theory of dynamic reliability is one of the most general approaches towards system reliability nowadays available, a small extension appears necessary to enable it to cover such situations. The reason is that the theory of dynamic reliability allows for the following:

• (Random changes between discrete system states at random time points.

• Deterministic changes in the physical variables of the system during the times in between.

An earthquake, however, may impose random changes on physical parameters of the system. This may be an impact on a physical variable in a narrow sense, like a sudden change in velocity of a movable mass due to acceleration, or it may be a change in a parameter, which is normally treated as a constant value, like an elasticity module, which changes due to plastic deformation. All such changes are immediate random changes of a continuous variable and depend, in general, from the intensity of the load. For an adequate modeling, an extension is suggested allowing for random changes of the physical variables during transitions between discrete states.

In the following, first, a formulation of dynamic reliability using frequency densities and the proposed extension are presented. Next, is presented a state model for the estimation of structural reliability. This framework is sufficient to handle a wide range of structural reliability problems. Finally, two specific problems are examined. The first is a simple example treated analytically, while the second is a realistic problem of structural reliability, solved with Monte-Carlo techniques under the proposed methodology.