Prandl established a universal velocity profile for flow parallel to the bed given by

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1 EM (Part VI) (g) The nderlayers shold be at least three thicknesses of the W 50 stone, bt never less than 0.3 m (Ahrens 98b). The thickness can be calclated sing Eqation VI-5-9 with a coefficient of 3 rather than. Becase a revetment is placed directly on the soil or fill material of the bank it protects, a single nderlayer also fnctions as a bedding layer or filter blanket. f. Blanket stability in crrent fields. Stone blankets constrcted of randomly-placed riprap or niformly sized stone are commonly sed to protect areas ssceptible to erosion by fast-flowing crrents. Blanket applications inclde lining the bottom and sloping sides of flow channels and armoring regions of tidal inlets where problematic scor has developed. Design of stable stone or riprap blankets is based on selecting stone sizes sch that the shear stress reqired to dislodge the stones is greater than the expected shear stress at the bottom developed by the crrent. () Bondary layer shear stress. (a) Prandl established a niversal velocity profile for flow parallel to the bed given by ' v ( κ ln y k s % B (VI-5-3) where κ = von Karman constant (= 0.4) y = elevation above the bed = velocity at elevation y k s = bondary roghness B = fnction of Reynolds nmber (= 8.5 for flly rogh, trblent flow) v * = shear velocity (= (τ o /ρ w ) / ) τ o = shear stress acting on the bed ρ w = density of water Eqation VI-5-3 can be expressed in terms of the mean flow velocity,, by integrating over the depth, i.e., ' v ( h m0 h dy ' v ( κ ln h k s % B & κ (VI-5-4) Fndamentals of Design VI-5-7

2 EM (Part VI) or v ( '.5 ln h k s (VI-5-5) when flly rogh trblent flow is assmed, which is sally the case for flow over stone blankets. Eqation VI-5-5 assmes niform bed roghness and crrents flowing over a distance sfficient to develop the logarithmic velocity profile over the entire water depth. (b) Bed roghness k s over a stone blanket is difficlt to qantify, bt it is sally taken to be proportional to a representative diameter d a of the blanket material, i.e., k s = C d a. Sbstitting for k s and v * in Eqation VI-5-5 and rearranging yields an eqation for shear stress given by τ o ' w w g.5 ln h C d a (VI-5-6) where w w = ρ w g is the specific weight of water. () Incipient motion of stone blankets. (a) Stone blankets are stable as long as the individal armor stones are able to resist the shear stresses developed by the crrents. Incipient motion on a horizontal bed can be estimated from Shield's diagram (Figre III-6-7) for niform flows. Flly rogh trblent flows occr at Reynolds nmbers where Shields parameter is essentially constant, i.e., Ψ ' τ (ρ a & ρ w ) gd a (VI-5-7) where τ = shear stress necessary to case incipient motion ρ a = density of armor stone Rearranging Eqation VI-5-7 and adding a factor K to accont for blankets placed on sloping channel side walls gives τ ' 0.04 K ( ) d a (VI-5-8) where is the specific weight of armor stone (= ρ a g), and VI-5-8 Fndamentals of Design

3 EM (Part VI) K ' & sin θ sin φ (VI-5-9) with θ = channel sidewall slope φ = angle of repose of blanket armor [. 40 o for riprap] (b) Eqating Eqations VI-5-6 and VI-5-8 gives an implicit eqation for the stable blanket diameter d a. However, by assming the logarithmic velocity profile can be approximated by a power crve of the form ln h h. C C d a d a β an explicit eqation is fond having the form d a h ' C T w w K gh (&β) (VI-5-30) where all the constants of proportionality have been inclded in C T. Eqation VI-5-30 implies that blanket armor stability is directly proportional to water depth and flow Frode nmber, and inversely proportional to the immersed specific weight of the armor material. The nknown constants, C T and β, have been empirically determined from laboratory and field data. (3) Stone blanket stability design eqation. (a) Stable stone or riprap blankets in crrent fields shold be designed sing the following eqation from Engineer Manal (Headqarters, U.S. Army Corps of Engineers 994). d 30 h ' S f C s w w K gh 5 (VI-5-3) where d 30 = stone or riprap size of which 30 percent is finer by weight S f = safety factor (minimm =.) to allow for debris impacts or other nknowns C s = stability coefficient for incipient motion = 0.30 for anglar stone = 0.38 for ronded stone Fndamentals of Design VI-5-9

4 EM (Part VI) (b) EM presents additional coefficients for channel bends and other sitations where riprap size mst be increased de to floccelerations. The methodology is also smmarized in Maynord (998). Eqation VI-5-3 is based on many large-scale model tests and available field data, and the exponent and coefficients were selected as a conservative envelope to most of the scatter in the stability data. Riprap stone sizes as specified by Eqation VI-5-3 are most sensitive the mean flow velocity, so good velocity estimates are needed for economical blanket designs. (c) Alternately, Eqation VI-5-3 can be rearranged for mean flow velocity to give the expression ' s f C s 5 h d 30 0 gk &w w w w d 30 (VI-5-3) (d) Eqation VI-5-3, which is similar to the well-known Isbash eqation, can be sed to determine the maximm mean velocity that can be resisted by riprap having d 30 of a given size. The main difference between Eqation VI-5-3 and the Isbash eqation is that the Isbash eqation mltiplies the term in sqare brackets by a constant whereas Eqation VI-5-3 mltiplies the sqare-bracketed term by a depth-dependent factor that arises from assming a shape for the bondary layer. The Isbash eqation is more conservative for most applications, bt it is still sed for fast flows in small water depths and in the vicinity of strctres sch as bridge abtments. (e) By assming the blanket stones are spheres having weight given by ' π 6 d 3 30 (VI-5-33) where is the stone weight for which 30 percent of stones are smaller by weight, Eqation VI-5-3 can be expressed in terms of stone weight as h 3 ' π 6 (S f C s ) 3 w w K gh 5 (VI-5-34) (4) Stone blanket gradation. (a) All graded stone distribtions (riprap) sed for stone blankets shold have distribtions conforming to the weight relationships given below in terms of or W 50 min (HQUSACE 994). W 50 min '.7 W 00 max ' 5 W 50 min ' 8.5 W 00 min ' W 50 min ' 3.4 (VI-5-35) (VI-5-36) (VI-5-37) W 50 max '.5 W 50 min '.6 (VI-5-38) VI-5-30 Fndamentals of Design

5 EM (Part VI) W 5 max ' 0.5 W 50 max ' 0.75 W 50 min '.3 (VI-5-39) W 5 min ' 0.3 W 50 min ' 0.5 (VI-5-40) (b) Recommended thickness of the blanket layer, r, depends on whether placement is sbmerged or in the dry as specified by the following formlas. (c) For blankets placed above water, the layer thickness shold be r '. W 50 min 3 '.5 3 (VI-5-4) with a minimm blanket thickness of 0.3 m. Blankets placed below water shold have layer thickness given by r ' 3. W 50 min 3 ' (VI-5-4) with a minimm blanket thickness of 0.5 m. VI-5-4. Vertical-Front Strctre Loading and Response a. Wave forces on vertical walls. () Wave-generated pressres on strctres are complicated fnctions of the wave conditions and geometry of the strctre. For this reason laboratory model tests shold be performed as part of the final design of important strctres. For preliminary designs the formlae presented in this section can be sed within the stated parameter limitations and with consideration of the ncertainties. Three different types of wave forces on vertical walls can be identified as shown in Figre VI (a) Nonbreaking waves: Waves do not trap an air pocket against the wall (Figre VI-5-57a). The pressre at the wall has a gentle variation in time and is almost in phase with the wave elevation. Wave loads of this type are called plsating or qasistatic loads becase the period is mch larger than the natral period of oscillation of the strctres. (For conventional caisson breakwaters the period is approximately one order of magnitde larger.) Conseqently, the wave load can be treated like a static load in stability calclations. Special considerations are reqired if the caisson is placed on fine soils where pore pressre may bild p, reslting in significant weakening of the soil. Fndamentals of Design VI-5-3

6 EM (Part VI) EXAMPLE PROBLEM VI-5- FIND: Riprap distribtion for a stable scor blanket over a nearly horizontal bottom GIVEN: The following information is known (English system nits shown in parentheses) Specific weight of riprap, = 5.9 kn/m 3 (65 lb/ft 3 ) Specific weight of water, w w = 0.05 kn/m 3 (64 lb/ft 3 ) Bottom slope, θ = 0 deg i.e., K =.0 Water depth, h = 6 m (9.7 ft) Depth-averaged mean velocity, =.5 m/s (8. ft/s) Stability coefficient, C s = 0.38 i.e., ronded stone Factor of safety, S f =. Gravitational acceleration, g = 9.8 m/s (3. ft/s ) SOLUTION: From Eqation VI-5-34 h 3 ' π 6 [(.) (0.38)] kn/m 3 [5.9&0.05] kn/m 3.5 m/s (.0)(9.8 m/s )(6 m) 5 '.54 (0) &6 The weight is fond as '.54 (0) &6 h 3 '.54 (0) &6 (5.9 kn/m 3 )(6m) 3 ' kn ' 8.6 N (.9 lb) The rest of the riprap distribtion is fond sing Eqations VI VI-5-40, i.e., W 50 max '.6 (8.6 N) '.4 N (5.0 lb) W 50 min '.7 (8.6 N) ' 4.6 N (3.3 lb) W 00 max ' 8.5 (8.6 N) ' 73. N (6.4 lb) W 00 min ' 3.4 (8.6 N) ' 9. N (6.6 lb) W 5 max '.3 (8.6 N) '. N (.5 lb) W 5 min ' 0.5 (8.6 N) ' 4.3 N (.0 lb) Blanket layer thickness for nderwater placement is fond sing Eqation VI-5-4 r ' kn 5.9 kn/m 3 3 ' 0.6 m (0.86 ft) The calclated vale for blanket thickness is less than the minimm vale, so se r = 0.5 m (.6 ft). VI-5-3 Fndamentals of Design

Two identical, flat, square plates are immersed in the flow with velocity U. Compare the drag forces experienced by the SHADED areas.

Two identical, flat, square plates are immersed in the flow with velocity U. Compare the drag forces experienced by the SHADED areas. Two identical flat sqare plates are immersed in the flow with velocity U. Compare the drag forces experienced by the SHAE areas. F > F A. A B F > F B. B A C. FA = FB. It depends on whether the bondary