# Bogdanoff’s Cumulative Damage Model

In this paper, a Bogdanoff’s cumulative damage (CD)-based model is used to predict the future condition and service life of reinforced concrete bridge decks. The proposed model assumes a probabilistic evolutionary structure of the damage accumulation process. The condition of the bridge deck is discretized into a finite state space with seven (7) damage states. A basic element of the model is the concept of duty cycle, which is a repetitive period of operation in the life of a deck in which the damage accumulation is assumed non-negative (Bogdanoff 1978). A duty cycle is defined as one-year in which the deck is subjected to de-icing salts in winter, freeze-thaw cycles, traffic load (in addition to its own weight).

The probability distribution of damage after a duty cycle is assumed to depend only on the duty cycle itself and the damage accumulated at the start of the duty cycle; thus its is assumed independent of how the damage was accumulated at the start of the duty cycle. This represents the first-order type of stochastic process correlation underlying the Markovian process (Bogdanoff 1978). These assumptions lead to the fact that the damage process can be modeled as a discrete-time and discrete – state Markovian process (Bogdanoff 1978). The probabilistic evolution of damage is completely determined by the transition matrix for each duty cycle and initial damage state. The transition probability matrix for a duty cycle is given by:

P=[pjk] j=1,2,…….. ,b; and k=1,2,….,b (1a)

with pjk= P(Dt+1=k I Dt=j) (1b)

where pjk represents the probability of the deck being in state k at the end of the duty cycle given it was in state j at the start of the duty cycle (with j < k for non-maintained systems). Damage states 1 to 6 are transient states, whereas damage state 7, denoted “state b” is called an absorbing state, which is a state that cannot be vacated without a maintenance action.

Given the adopted condition rating scale and short duration of the duty cycle or transition time (1 year), the probability of deteriorating by more than one state (i. e. multiple damage states transitions) may be assumed negligible (Golabi and Shepard 1997; Lounis 2000; Morcous et al. 2003). Therefore, the transition matrix is greatly simplified and has only two elements per row, namely pkk and pkk+1, which is referred to as the “unit jump Markov chain” (Bogdanoff 1978). Given the uncertainty in defining the end of life or failure criterion, it is possible to have different definitions of the absorbing state, depending on the requirements of the bridge owner, risk of failure, etc. The initial state of damage D0 is identified by the vector p0=[po(i)].=k b, where po(i) is the probability of being in state i at time t=0. This initial damage may arise from poor materials, inadequate design and/or construction. It follows from Markov chain theory (Bogdanoff 1978; Ross 1996) that the damage state vector at time t, pt, is given by:

Pt= Po PP2……… P, = [pt(1) pt(2) pm (2a)

where P; is the transition matrix for the jth duty cycle, and pt(k) is the probability of being in state k at time t. If we assume that the duty cycles are all of constant magnitude throughout the deck lifetime, then the transition probability matrices are time-invariant and equal to P, which yields a stationary stochastic process. Therefore, Eq. (2a) simplifies to:

Pt= Po P (2b)

The above transition probability matrix is generated from the data collected during the inspections of the bridge decks. Contrary to lifetime models, the transition matrix and thus the proposed cumulative damage model can be developed from a limited set of data, which then can be further refined using the Bayesian updating approach (Golabi and Shepard 1997; Lounis and Madanat 2002). The probability that the deck be in damage state j at time t is given by:

P(Dt = j)= pt(j) (3a)

The cumulative distribution function of damage at time t, Dt, is defined by:

FDt(j)= P(Dt < j) = £ P, (k) (3b)

k=1

The expected damage at time t, E[D(t)] is given by:

b

E[D(t)]= ^ jpt(j) (3c)

і=i

The service life (L) of the deck may be defined as the time to absorption at state b. For the case of an initial damage vector with p0(1)=1, its cumulative distribution function FL is given by (Bogdanoff 1978):

Fl® = P(L < t) = Pt(b) t=1, 2,…., n (4)

The expected service life E[L] is given by the mean time to absorption (Bogdanoff 1978), i. e.:

TO

E(L) = ^ [1 – FL(t)] (5)

t=1

For the case of an initial damage vector with multiple nonzero elements, the cumulative distribution function of service life (or time to absorption) is given by (Bogdanoff 1978):

b-1

FT(t) = P(T < t) = £ p0(k)Fn(t) (6)

к =1

where FTk(t)is the cumulative distribution function of the time at which the damage state first enters the absorbing state, given the initial damage state is k.

The probabilistic prediction of accumulation of damage in the bridge deck using Eq.(2a) is illustrated in Fig.2, which indicates the evolution with time of the probability mass function of the damage. In Fig.2, it is seen that in the early stages of the deck life, the probability mass of the damage is near state 1, but with aging and damage accumulation, this probability mass shifts to higher damage states. Ultimately, if no maintenance is undertaken, all the probability mass accumulates in the absorbing state 7 or state “b”.

 A Fig. 2. Probabilistic evolution of bridge deck deterioration using Bogdanoff’s CD model