Kramer’s modulus

III) Kramer s Modulus—Kramer (1935), in studies on tractive forces involved in model channels, developed an expression for measuring uniformity which depends on the ratio of the areas above and below the 50 percent line, as indicated in Figure 13. Kramer s uniformity modulus is the ratio of the area below to the area above this line. The modu-... 

Figurb 13. Schematic-summation Curve Showing Method of Determining Kramer s Modulus.

The modulus is also said to be a measure of the voids ratio. It has a maximum value of 1 for uniform material, and smaller values if finer or coarser materials are added. When the voids between the large particles are filled with smaller particles the modulus decreases. However, it is hard to conceive that Kramer s modulus is any more suited to such representation than the standard deviation. 

Mean Velocity, and Manning s n for an Experimental Flume. Average Diameter of Sand 0.347 mm, Kramer s Modulus 0.643, and Slope 0.0015. 

Calculate the critical tractive force on a flume bed of sand whose average diameter is 0.033 in. and Kramer s modulus 0.44. Take the specific gravity of sand as 2.65. 

Work of U. S. Waterways Experiment Station—The most comprehensive study of silt movement in recent years is that by the Corps of Engineers (1935a), published by the U. S. Waterways Experiment Station. To some extent this study followed the pattern used by Kramer, but included more different kinds of material. The mean sizes for eight of the sands composing the beds ranged from 0.205 to 0.586 mm for one other sand the mean size was 4.08 mm. The sand, having a mean diameter of 0.586 mm, had a uniformity modulus of 0.280 all the others had moduli of approximately 0.5. The critical tractive-force equation obtained was... 

Hvidt S, Kramer O, Batsberg W, Ferry JD (1980) Contribution of entanglements to the equilibrium modulus of 1,2-polybutadiene networks at small strains and estimate of the fi"ont factor. Macromolecules 13(4) 933-939... 

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