The forces acting in the plane of the side slope are Fd and Ws sinθ as shown in Figure 6.13b.
The angle θ is the side slope angle. The lift force acts normal to the side slope and the
component of submerged weight Ws sinθ acts normal to the side slope as shown in Figure
6.13c.
At incipient motion, there is a balance of moments about the point of rotation such that
e2 Ws cos θ = e1Ws sin θ cos β + e3Fd cos δ + e4Fl
(6.1)
The moment arms e1, e2, e3 and e4 are defined in Figure 6.13c and the angles δ and β are
defined in Figure 6.13b.
The stability factor S.F. against rotation of the particle is defined as the ratio of the moments
resisting particle rotation out of the bank to the submerged weight and fluid force moments
tending to rotate the particle out of its resting position. Accordingly,
e2 Ws cos θ
S.F. =
(6.2)
e1Ws sin θ cos β + e3Fd cos δ + e4Fl
The following particle stability analysis was first derived by Stevens (1968). The analysis is also
presented in Simons and Senturk (1992). This analysis shows that the stability factor for rock
riprap on side slopes where the flow has a non-horizontal velocity vector is related to properties
of the rock, side slope and flow by the following equations:
cos θ tan φ
S.F. =
(6.3)
η' tan φ + sin θ cos β
in which
cos λ
-1
β = tan
(6.4)
2 sin θ
+ sin λ
η tan φ
21τo
η=
(6.5)
(S s - 1) γD s
and
1 + sin (λ + β)
η′ = η
(6.6)
2
Given a rock size Ds, of specific weight Ss and angle of repose φ and given a velocity field at an
angle λ to the horizontal producing a tractive force τo on the side slope of angle θ, the set of
four equations (Equations 6.3, 6.4, 6.5, and 6.6) can be solved to obtain the stability factor S.F.
If S.F. is greater than unity, the riprap is stable; if S.F. is unity, the rock is at the condition of
6.22
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