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217 acb coefficient, Equation 7-1 b flow parallel wing-wall abutment thickness, Figure 6-9 B upstream width of the flume, Equation 5-19 Bf floodplain width, Figure 7-3 B1 original channel width, Equations 5-4 and 5-6 in Table 5-2 B2 constricted channel width, Equations 5-4 and 5-6 in Table 5-2 C coefficient, Equation 5-18 Cs stability coefficient for incipient failure: 0.3 angular rock, 0.375 rounded rock, Equation 5-14 CT blanket thickness coefficient, given by Figure 1 in Maynord 1993, Equation 5-14 Cv vertical velocity distribution coefficient, Equa- tion 5-14 C* coefficient determined from laboratory and field testing, Equation 5-15 d sediment diameter, Equation 8-7 dabut,avg time-averaged scour depth at abutment, Table 9-2 db distance between the average bed level and the bottom of the apron, Figure 7-13 dc maximum scour at the countermeasure, Table 9-3 dcm,avg time-averaged scour depth at the countermea- sure, Table 9-4 dmax,abut maximum scour depth at the abutment foun- dation, Table 9-11 dmax,abut,inst maximum instantaneous scour depth at abut- ment, Table 9-4 dmax.col maximum scour near the abutment collar countermeasure, Table 9-11 dmax,sp1,avg maximum scour depth behind the first spur dike, Table 9-9 dmax,sp2,avg maximum scour depth behind the second spur dike, Table 9-9 dmax,sp3,avg maximum scour depth behind the third spur dike, Table 9-9 dmax,sp1,inst maximum instantaneous scour depth in front of the first spur dike dmax,sp2,inst maximum instantaneous scour depth at the second spur dike, Table 9-10 dabut,avg time-averaged scour depth at the abutment, Table 9-2 dmax,ch scour depth in the channel away from the abutment, Table 9-1 dmax,cm,inst maximum instantaneous scour depth at the countermeasure, Table 9-4 dmax,dn,abut scour depth at a short distance downstream of the downstream corner of the abutment, Table 9-1 dmax,sp,avg maximum scour depth at the spur dike, Table 9-8 dmax,up,abut scour depth at the upstream corner of the abutment, Table 9-1 dM tail-water depth immediately downstream of the scour hole, Equation 5-8 dsA scour reduction at the abutment with scour countermeasure, Section 6.4.1 dsAO scour depth at the abutment without scour countermeasure, Section 6.4.1 dsmax maximum scour depth, Section 7.3.4 dsO equilibrium scour depth below the bed surface without countermeasures, Section 8.5.3 ds1 vertical distance from the original top of the apron to the apron settlement near the abut- ment, Figure 7-13 ds2 vertical distance from the top of the apron to the apron edge settlement, Figure 7-13 dsf depth of the scour hole relative to the flood- plain, Figure 8-12 Notation
d16 particle size of which 16 percent of the grains are finer, Section 7.2.1 d50 median particle size, Section 6.3.1 d84 particle size of which 84 percent of the grains are finer, Section 7.2.1 D riprap diameter, Table 5-6 DB geobag thickness, Equation 5-23 Dn design diameter of the cable-tied blocks, Equation 5-20 DR equivalent riprap diameter, Equation 7-17 Ds distance between the farthest spur dike tip at the main channel end and the abutment tip, Table 9-8; and spacing between spur dikes or the spur dike and the abutment, Table 9-9 D30 riprap size for which 30 percent by weight are finer, Equation 5-13 D50 median grain or riprap size, Equations 5-6 in Table 5-2 E parameter that has a value of 0.86 for loosely placed stones in flowing water and 1.2 for those that have become embedded, Equation 5-10 Ev kinematic eddy viscosity, Section 8.4.2 f Lacey silt factor, Equation 5-1 in Table 5-2 Fbo Blenchâs zero bed factor, which is a function of grain size, Equation 5-2 in Table 5-2 Fn Froude number, Equations 5-4 and 5-5 in Table 5-2 Fr flow Froude number, Equation 5-15 Fr2 Froude number in the contracted section, Equations 5-F through 5-H in Table 5-5 h1 main channel bank height, Table 9-1 Ha flow parallel thickness of abutment, Figure 6-3 Hb minimum required block height, Equation 7-2 Ht total drop in head, measured from the upstream to downstream energy grade line m, Equation 5-8 Hcb height of concrete blocks, Equation 5-22 k function of approach conditions, Equations 5- 1 through 5-5 in Table 5-2 ks roughness height, Equation 8-2 K function of drag coefficient (CD) that varies between 2.5 and 5.0, Equation 5-4 in Table 5-2 K1 side slope correction factor, Equation 5-14 Kd slope factor, Equation 5-17 Kh depth parameter, Equation 5-17 Ks shape factor, Equations 5-G and 5-H in Table 5-5 KS shape factor associated with the abutment shape, Equation 7-17 Ksl embankment slope factor, Equations 5-B and 5-C in Table 5-5 KT turbulence adjustment factor, Equation 5-17 La abutment length perpendicular to flow, Equa- tion 5-19 Lb bottom length of cable-tied block, Figure 7-8 Lp pier length, Section 5.7; and projected width of the parallel-wall countermeasure, Figure 9-14 Ls apron spread length, Figure 7-28 Lsd effective length of the spur dike, Equations 5- 5 and 5-7 in Table 5-2 Lsdp spur dike protrusion length, Table 9-8 Lspw wall length, Table 9-4 Lt top length of cable-tied block, Figure 7-8 n Manning coefficient, Equation 7-5 Nsc dimensionless stability factor for riprap stone, Equation 5-10 Nsd number of spur dikes, Table 9-9 pm volume fraction pore space within the mat- tress, Equation 7-2 Pb protrusion of the blocks above bed level, Equation 7-4 Pe Peclet number, Section 8.4.2 q discharge per unit width, Equations 5-2 and 5-3 in Table 5-2 Q total discharge, Equation 5-1 in Table 5-2 Qf lateral or floodplain flow discharge, Section 9.3.1 QO flow directly upstream of the bridge opening, Figure 8-26 QT total flow in the compound channel upstream of the bridge crossing, Figure 8-26 Q100 discharge in the 100 feet of stream adjacent to the abutment, Section 9.3.1 ra apron width, Figure 6-9 rs assumed multiple of scour at a dike taken as 11.5 by Laursen, Equation 5-7 in Table 5-2 rt radius of the spill-through abutment toe, Equation 5-19 Rb centerline radius of curvature of bend, Equa- tion 5-14 Rdmax distance to the deepest point of the scour hole from the abutment end, Equation 8-9 R50 median grain size of stone that makes up the grade control, weir, or check-dam, Equa- tion 5-9 Sb parallel-wall countermeasure side slope, Table 9-5 Scb specific gravity of the blocks, Equation 7-5 Sfa stability factor varying from 1.6 to 2.0 for abut- ment protection, Equation 5-C in Table 5-5 218
Sf safety factor >1, Equation 5-14 Sn parallel-wall countermeasure end slope, Table 9-5 Sr the specific gravity of the riprap stones, Equa- tion 5-11 SS specific gravity of riprap stone, Equation 5-20 SSB specific gravity of the geobag, Equation 5-23 t run time, Table 9-2 te experiment run time, Table 9-1 tp thickness of the protection unit, Equation 5-17 TI turbulent intensity at 10 percent of the water depth above the bed, Equation 5-20 U mean flow velocity, Figure 5-34 u* shear velocity, Section 6.3.1 u*c critical value of the shear velocity associated with bed-particle movement, Section 6.3.1 V local depth-averaged velocity, Equation 5-14; and cross-sectionally averaged velocity, Section 7.2.1 Vb velocity at abutment end, Equation 5-B in Table 5-5; and velocity at 10 percent of the water depth above the bed, Equation 5-20 Vc critical velocity for bed sediment movement, Figure 8-7 Vcr critical threshold velocity for stone, Equation 5-11 Vcs critical velocity for bed sediment movement, Section 5.7 Vr velocity at a level of one-rock diameter above the bed, Equation 5-A in Table 5-5 Vtip velocity just outside the separation zone at the end of the abutment, Table 8-4 V2 mean velocity in contracted bridge section, Equations 5-D and 5-E in Table 5-5 V2-ave bridge section depth-averaged flow velocities, Table 7-6 V2-surf upstream and bridge section surface flow velocities, Table 7-6 V0 mean velocity, Section 6.3.1 W water surface width at upstream end of bend, Equation 5-14 Wa apron width, Equation 5-19 WCR critical block weight, Figure 5-34 Wmin width of apron in the unsettled region, Figure 7-13 Wo apron extent, Equation 8-16 x streamwise direction, Figure 7-3 y transverse direction, Figure 7-3; flow depth, Equation 5-12; and average depth in uncon- stricted section, Equations 5-1 and 5-4 through 5-7 in Table 5-2 yc critical depth of flow, Equation 5-9 yf floodplain flow depth, Figure 7-3 ym main channel flow depth, Figure 7-3 ys equilibrium scour depth measured from water surface, Equations 5-1 through 5-7 in Table 5-2 y2 flow depth in contracted section, Equations 5- D, 5-G; and 5-H in Table 5-5 Y flow depth in the bridge section, Equation 7-17 Y0 flow depth, Section 6.3.1 z vertical direction, Equation 8-2 %max,abut maximum percentage of scour reduction at the abutment, Table 9-9 %max,abut,avg percentage reduction in time-averaged scour depth at abutment, Table 9-4 %max,abut,inst percentage reduction in maximum instanta- neous scour depth at abutment, Table 9-4 b angle of the boundary on which the geobag is placed, Equation 5-24 e horizontal distance from floodplain wall to opposite edge of scour hole, Section 8.1.4 r angle of sediment repose, Section 6.3.1 s main channel bank slope angle, Equation 8-8 x horizontal distance from the abutment tip to the deepest scour location in the flow direc- tion, Figure 8-12 y horizontal distance from the abutment tip to the deepest scour location in the transverse direction, Figure 8-12 2 horizontal width of the apron, including set- tled and unsettled portions, Figure 7-13 angle with the horizontal of the settled portion of apron, Figure 7-13 s unit weight of stone, Equation 5-14 w unit weight of water, Equation 5-14 D bed-form trough depth, Figure 7-14 H bed-form trough height, Figure 7-14 H-max maximum bed-form height, Equation 7-3 L bed-form trough length, Figure 7-14 L-ave average bed-form length, Equation 7-3 c stability factor for current, Equation 5-17 change in a quantity, Equation 8-1 m relative density of protection system, Equation 5-17 weight per unit area of the mattress, Equation 5-21 stability factor, Equation 5-A in Table 5-5 spur angle, Figure 5-5 r angle of repose, Equation 5-B in Table 5-5 C critical value of the Shields parameter for particle geobag entrainment, Equation 5-23; and angle of repose of the sediment forming the boundary, Equation 5-24 219
fluid kinematic viscosity, Equation 8-3 cb density of the concrete blocks, Equation 5-21 g standard deviation of sediment size, Section 6.3.1 bed shear stress, Equation 8-6 c critical bed shear stress for incipient move- ment of bed sediment, Equation 8-7 wc critical bed shear tress on local bed slope, Equation 8-8 stability parameter, Equation 7-18 ai downstream apron initiation angle, Equation 5-19 sl slope angle, Equation 5-B in Table 5-5 st stability parameter, Equation 5-23 cr critical shear stress parameter, Equation 5-17 repose angle of the bed material, Equation 8-8 vorticity, Equation 8-1 220
