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~ a 9 ] 3 ~ 9 I I I ~ g I I I I I I I 39 Equilibrium Slope The Meyer-Peter and Muller bed-load equation is one of many mathe- matical expressions used to analyze and predict the movement of noncoheslve sediment in rlvers. It is selected here because of its simplicity and not foranyadd1tlonalintrfns1cvaluefthasoverotherequatlons. The equation was developed from flume experiments with water depths ranging from 0.3 to 4 feet and slopes from 0.0004 to 0.02. The mean size of the sediments were varied from O.4,ml11imeters (medium sand) to 30 millimeters (coarse gravel). It has been criticized as predicting too much sediment transport in gravel bed rivers. For situations in which no bed forms occur in essentially one- dimensional uniform flow, the Meyer-Peter and Muller equation can be written " q.= ass {.9.)l/2[01l3 q3/5i11/10s7/l0_0,047(S _ ~ (Ss _ I) y . Y -m s lh~]3/2 (1) in which qs = transport rate of bed material per unit width of channel, lb/sfft 5, . specific weight of the sediment 9 . acceleration due to gravity, ft/s2 , . unit weight of water, lb/ft3 q ~ water discharge rate per unit width of channel, ft3/s/ft ~ = effective diameter of the sediment, ft S = riverbed slope. To arrive at this expression, use has been made of the Manning equation 1.486 y513 Sl/2 , (2) q. in which y '" uniform depth of flow, ft n = Manning's roughness coefficient end the roughness predictor n"0.039SC\nl/6 (3) Under conditions of uniform steady flow of water and sediment in a one~dimens1onal channel, the equilibrium slope is then 40 I I I I I I I I I I I I I I I I I I I 5 . Ss - 1 2/3 Il/3 2/3 [f ass) (g) qs +0.047(Ss 0.113yq3/5dn,1/l0 (,) 10/7 1h<\.] When the bed load transport is large, the term 0.047 (Ss - l)ydm is small in comparison with the other term in the numerator of Equation~. Con- versely, when the bed transport is small (qs ~ O), Equation 4 reduces to S=0.286(Ss_l)10/7dn,9/7q-6/7 (5) which is the expression for the bed slope of a straight gravel bed channel with no bed forms and uniform constant flow. In order to solve the degradation problem mathematically, one other relation is needed. that being how the bed material size changes as the bed degrades. For the South Platte River in the study reach, it is generally found that the material underlying the riverbed is some combination of sand, gravel and cobbles. Ho shale outcrops were observed in the reaches upstream or downstream from the channelized reaches. These alluvial materials were transportedina larger llnd different river in past time and are not un i- formly distributed. Knowledge of the sizes of these underlying materials is scanty. The logs of the holes drilled for the exploration of bridge foundations state merely what materials were present but not their size distributions. Because of the probable lack of spatial uniformity in underlying bed material, the lack of any samples from below the surface for which the size distributions are known and the non-uniformity of the material on the sur- face of the bed, no conclusive value of the amount of degradation antici~ pated in the South Platte Ri~er can be drawn at this time. However, the equations were employed to study the range of possible conditions which could arise. The findings are as follows: 1. Shown in Figure 10. the bed material samples taken from the South Platte Ri~er range in size from medium sand to very coarse gravel. The average of all these samples is a very fine gravel which according to the method of Gessler (1971). will degrade and anmor under conditions predicted for the future. The median diameter of the armor coat would increase from 2.2 millimeters to approximately 15 millimeters under an applied shearing stress of 0.4 pounds per square foot. This stress would occur when the unit discharge is on the order of 25 cubic feet per second per foot of channel width. 2. The computed size of the bed material needed to form a stable armor coat on the riverbed for a design discharge of 33.000 cubic feet per second in a 2SD-foot wide channel is in the order of 3.5 inches in diameter. This size exists beneath the ri~er bed.