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<br />Shear Stress Related to Hydraulic Radius and Gradient <br /> <br />Water flows in a channel as a function of gravity and develops forces that <br />act in many directions depending on the amount of turbulence, but with the <br />primary vector in the direction of flow. Friction forces act in the opposite <br />direction to the flow and, when considered over an area, are referred to as <br />boundary shear. The forces acting on a body of water are shown in figure 3. <br />For uniform flow, the pressure forces are equal and act in opposite directions, <br />and all accelerations are ~ero. Analyzing the flow condition shown in figure 3 <br />and converting forces to shear stresses gives the equation: <br /> <br />toPaL = yAaL sin e <br /> <br />(1) <br /> <br />For <br />the <br /> <br />small angles, sin e = tan e = So. <br />average shear stress on the boundary <br /> <br />By dividing by Pal and substituting So, <br />is given by: <br /> <br />to <br /> <br />= yRSo <br /> <br />(2) <br /> <br />where to = mean boundary shear acting over the wetted perimeter <br /> P = wetted perimeter <br /> L = length <br /> y = unit weight of water <br /> A = cross-sectional area <br /> e = angle of bed slope <br /> R = hydraulic radius <br /> So - channe 1 bed slope <br /> <br />F2 <br /> <br /> <br />,/ <br />A <br />,/ <br /> <br />~L <br /> <br />--- <br /> <br /> <br />/ <br />~ <br />/ <br /> <br />I <br /> <br />\-----t------ <br /> <br />\ I <br />\ I <br /> <br />\. ,---- <br />\ ,..- <br />,,>L--- Wsin <br />~~; ------------ <br /> <br />----- <br />------ <br />---- <br /> <br /> <br />\ <br />\ <br />\ <br />\: <br />\ <br />--+ <br />\ <br /> <br />F1 <br /> <br />Note: dm Is measured in the vertical and is a close <br />approximation (0.5 percent) as dm =dm cose <br />for channel slopes less than 0,1. <br /> <br />F1' F2 = Forces of static pressure <br /> <br />W=Welght of water <br /> <br />FIGURE 3, Three.dimensional free body diagram of forces acting on a water mass, <br /> <br />7 <br />