Hiroaki Nishi

Optimal control of maintenance strategies including repairs and reinforcements for concrete tunnel lining

for ICOSSAR 2013 Optimal control of maintenance strategies including repairs and reinforcements for concrete tunnel lining Hiroaki Tanaka, Atsushi Sutoh, Osamu Maruyama, Takashi Satoh and Hiroaki Nishi Dept. of Applied Analysis and Complex Dynamical Systems, Kyoto University, Kyoto, Japan Dept. of Urban and Civil Engineering, Tokyo City University, Tokyo, Japan Civil Engineering Research Institute for Cold Region, Sapporo, Japan Correspondence: hiroaki@i.kyoto-u.ac.jp 1. Purpose of this study In this paper, we theoretically discuss an optimal maintenance strategy for concrete tunnel lining, based upon a new probabilistic model describing random growth of the quantified damage in the concrete lining. Our probabilistic model is based upon a random differential equation driven by a Poisson white noise. The theory of stochastic control is applied to derive the optimal strategy that is given as the solution of the HJB (Hamilton-Jacobi-Bellman) equation. 2. Probabilistic model for damage growth Let X(t) be a quantified damage in an objective concrete tunnel lining at time t. We here suppose that its temporal variation is described by the following stochastic differential equation; dX(t) = μX(t)dt+X(t−)dC(t), (1) where μ is a positive constant, C(t) is a compound Poisson process and X(t−) is a leftcontinuous version of X(t). The first term in Eq.(1) represents usual growth of damage with small scatter and the second term represents unusual random growth of damage. We assume that “failure”of the concrete lining occurs when the damage X(t) exceeds a critical value xc. Once such a failure occurs, we suppose that a large scale “reinforcement”is immediately performed. 3. Basic scheme of repair and reinforcement In this paper, we basically suppose the following maintenance strategy; • The maintenance program stars at t = 0[year] and terminates at t = T [year], which is determined in advance. • Suppose that the damage value is known to be x by an inspection performed at time t. If x is less than a certain prespecified threshold value, denoted by xth (0 < xth ≤ xc), we execute a “repair”of the damage so that the damage value is reduced from x to x − a(t), where a(t) is a control variable in our optimization procedure. On the other hand, if xth ≤ x ≤ xc, a large scale reinforcement is performed. A cost required for such a maintenance procedure is expressed as R1(t, x, a(t)), where R1(t, x, a) is a given deterministic function.