PSI - Issue 5

H Hammoum et al. / Procedia Structural Integrity 5 (2017) 69–76 H Hammoum et al./ Structural Integrity Procedia 00 (2017) 000 – 000

71 3

P f = ∫ f x (x)dx D f

(3)

D f is the failure domain defined by: D f = {X ∈ R/G(X) ≤ O} f x (x) is the probability density function of the random vector {X} constituted by the random variables x i , whose realizations are {X} = {x 1 , x 2 , x 3 … . . x n } t . 3. Failure modes and functions of limit states of an open channel Traditionally, the design of Reno mattress of open canals is based on a deterministic analysis. Its calculation consists first on the verification of the canal bottom and the canal banks stability in terms of traction forces. The second verification is the stability of a revetment with reference to water velocity. The deformation effects are estimated at the third step. The residual velocity at the underside of the revetment is evaluated at the last step. The interested reader by further details of this deterministic method can consult the following reference (Agostini et al, 2012). A stone revetment is considered to be stable when there is no movement of the individual stones. This holds good for Reno mattress and gabion revetments in which the stone is encased in steel wire mesh. The limit of the revetment's stability occurs at the point at which the stones are about to move, i.e. the condition of "initial movement". The revetment is stable when the shear stress asserted on the revetment is less than the shear stress at the critical condition of "initial movement", reached without stone movement, this state is given by the following relation: (4) We can define the ultimate limit state function G 1 of exceeding stability of the bottom of the canal as follow: 1 b c G : -   (5) b c τ τ  As for the bottom of the canal, the revetment is stable when the shear stress asserted on the revetment is less than the shear stress at the critical condition reached without stone movement. This state is given by the following relation: m τ τ s  (6) For the banks of a trapezoidal cross-section channel, the shear stress can be expressed as a function of the water depth and the canal slope. The critical shear stress can be expressed as a function of the bank side slope and the angle of internal friction of the stone fill. We can define the ultimate limit state function G 2 of exceeding stability of the canal banks as follow: 2 m s G : -   (7) 3.3. Verification in terms of velocity It is accepted in practice to evaluate the stability of a revetment with reference to water velocity. The average velocity noted V of the current section is expressed as a function of the water depth and the canal slope. Critical velocity noted V C for a given revetment depends also on the water depth, the Froude number and the mattress 3.1. Verification in terms of traction forces in the canal bottom 3.2. Verification in terms of traction forces in the canal banks