#
Upper Bound for Two-Sided Barrier Problem

*by* Arnold L. Sweet,

**Serial Information**:

*Journal of the Engineering Mechanics Division*, 1969, Vol. 95, Issue 6, Pg. 1369-1378

**Document Type:** Journal Paper

**Abstract:** An upper bound is derived for the probability that a random process

X (t) will take values outside an interval [ – λ

_{1} (

t), λ

_{2} (

t) ] for 0 ≤ t ≤ T, in which λ

_{1} (

t) and λ

_{2} (

t) are nonnegative continuously differentiable curves. The random process is assumed to be separable, have mean zero, and to be mean square differentiable at least once. The upper bound is shown to be less than one previously obtained by Shinozuka and Yao. Under the additional assumption that

X(

t) is normal, expressions for the upper bound are found, and are shown to depend on the joint probability density of

X(0),

X(

t), and

X(

t). In order to evaluate the expressions, the integral of a bivariate normal distribution must be evaluated. The results of computations for symmetric barriers [ λ

_{1} (

t) = λ

_{2} (

t) ] are shown, and conditions under which significant differences between the new upper bound and that previously derived can be expected are indicated. It is shown that significant improvements may result when stationary rather than transient random processes are considered.

**Subject Headings:** Curvature |

Joints |

Integrals |

Computing in civil engineering |

Symmetry |

Stationary processes |

Transient response |

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