Ch. 31 TRACTIVE FORCE 69 



a movable bed. This friction will determine the relationship between 

 stage and discharge in the river, and it will determine the average 

 velocity and the velocity distribution. It has been found that, basi- 

 cally, this friction may be described by von Karman's logarithmic 

 equations, using the constants which Keulegan (1938) has derived 

 from Nikuradse's experiments for the flow along rough walls. The 

 roughness diameter k in these relationships is represented by the 

 grain diameter in the case of sediment of uniform size. For sediment 

 mixtures the diameter used in computing the roughness factor k is 

 the diameter known to engineers as D c> 5. That is, 65 percent of the 

 weight of the sediment is composed of particles smaller than D 65 and 

 35 percent of particles larger than this diameter. 



Additional flow resistance is caused by the ripples and bars of the 

 sediment bed and by the irregularities of the channel. Both may be 

 expressed in terms of the specific sediment rate. From flume studies 

 where channel irregularities are practically eliminated, one can learn 

 that the additional friction due to ripples and bars is small for the 

 very lowest and for extremely high rates of sediment transport, reach- 

 ing a maximum value at an intermediate rate where sediment bars 

 are most commonly developed. In natural rivers not constricted by 

 artificial banks, the reduction of the additional friction at low sedi- 

 ment rates is counteracted by channel irregularities such that the 

 overall friction coefficient is highest at lowest flows. These relation- 

 ships are not sufficiently understood yet but are just as important for 

 the calculation of an alluvial river as is the bed-load equation itself. 



The above discussion pertains primarily to the rate of sediment 

 transportation. Another phase of the sediment problem that has been 

 investigated extensively in the past is the critical condition control- 

 ling the commencement of movement of the bed material. 



TRACTIVE FORCE 



The most common conception of the mechanics of bed-load move- 

 ment is that a dragging force is exerted on the bed of a stream by the 

 flowing water. This force, termed the tractive force, is the dragging 

 or entraining force exerted at the base of a prism of water of unit 

 area of the bed and of height equal to the water depth sliding, under 

 the influence of gravity, down an inclined plane having a given slope. 

 Critical tractive force is that tractive force which creates "general 

 movement" of the bed material. General movement, as commonly 

 used in this sense, is that condition under which sand grains up to and 

 including the largest size available are in motion. This critical con- 



