The reference velocity Vr = 1.83 m/s, and the embankment side slope angle θ = 18.4 which
corresponds to a 3:1 side slope. The velocity vector angle with the horizontal λ = 20. If the
embankment is covered with dumped rock having a specific weight Ss = 2.65 and an effective
rock size Dm = 0.305 m, determine the stability factor.
From Equation 6.16
(
)
0.30 Vr2
(0.30) 1.83)2
η=
=
= 0.203
(S s - 1) gDm (2.65 - 1) (9.81) (0.305)
This dumped rock has an angle of repose of approximately 35 according to Figure 3.4.
Therefore, from Equation 6.4.
cos λ
cos 20
-1
-1
β = tan
= tan
= 11
2 sin θ + sin λ
2 sin 18.4
+ sin 20
η tan φ
0.203 tan 35
and from Equation 6.6
1 + sin (λ + β)
1 + sin (20 + 11)
η′ = η
= 0.154
= 0.203
2
2
The stability factor for the rock is given by Equation 6.3
cos θ tan φ
cos 18.4 tan 35
S.F. =
=
= 1.59
n′ tan φ + sin θ cos β 0.154 tan 35 + sin 18.4 cos 11
Thus, with a stability factor of 1.59, this rock is more than adequate to withstand the flow
velocity.
By repeating the above calculations over the range of interest for Dm (with φ = 35), the curve
given in Figure 6.33 is obtained. This curve shows that the incipient motion rock size is
approximately 0.107 m and the maximum stability factor is less than 2.0 on the 3:1 side slope.
The stability factor of a particular side slope riprap design can be increased by decreasing the
side slope angle θ. If the side slope angle is decreased to zero degrees, then Equation 6.3 is
applicable and
1
1
S.F. =
=
= 4.93
η 0.203
The curve in Figure 6.34 relates the stability factor and side slope angle of the embankment
(for λ = 20, Dm = .0305 m ft and Vr = 1.83 m/s). The curve is obtained by employing
Equations 6.3, 6.4, 6.16, and 6.6 for various values of θ.
6.58
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