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<br />(a) Spiral lransition curves. For channels in which <br />surface disturbances need to be minimized. spiral <br />transition curves should be used. The gradual increase in <br />wall deflection angles of these curves results in minimum <br />wave heights. Two spiral curves are provided. one <br />upStre:lm and one downstteam of the cenlra! circular <br />curve. The minimum length of spirals for unbanked <br />curves should be detennined by (see Douma. p 392. in <br />Ippen and Dawson 1951) <br /> <br />Ls = 1.82 ,,: <br /> <br />(2-32) <br /> <br />where y is the sttaight channel flow depth. <br /> <br />(b) Spiral-banked curves. For rectangular channels. <br />the invert should be banked by rotating the bottom in <br />transverse sections about the channel center line. Spirals <br />are used upStre:lm and downstre:lm of the cenlra! curve <br />with the banking being accomplished gradually over the <br />length of the spiral. The maximum amount of banking or <br />difference between inside and outside invert elevations in <br />the circular curve is equal to twice the superelevation <br />given by Equation 2-31. The invert along the inside wall <br />is depressed by !>y below the center-line elevation and <br />the invert along the oUlSide wall is raised by a like <br />amounL Wall heighlS are usually designed to be equal on <br />both sides of the banked curves and no allowance needs <br />to be made for superelevation aroWld the curve. The <br />minimum length of spiral should be 30 times the amount <br />of superelevation (!>y) (USAED, Los Angeles. 1950). <br /> <br />Ls = 30!>y <br /> <br />(2-33 ) <br /> <br />The detailed design of spiral curves is given in <br />Appendix D. A computer program for superelevation and <br />curve layout is included. Banked inverts are not used in <br />trapezoidal channels because of design complexities and <br />because it is more economical to provide additional free- <br />board for the moderate amount of superelevation that <br />usually occurs in this type of channel. <br /> <br />c. Limiting curvature. Laboratory experiments and <br />field experience have demonslI:lled that the helicoidal <br />flow. velocity distribution distortion. and separation <br />around curves can be minimized by properly proportion- <br />ing channel curvarure. Woodward (1920) recommends <br />that the curve I':ldius be greater than 2.5 times the channel <br /> <br />EM 1110-2-1601 <br />1 Jul 91 <br /> <br />width. From experiments by Shukry (1950) the radius of <br />curvature should be equal to or greater than 3.0 times the <br />channel width to minimize helicoidal flow. <br /> <br />(1) TGII1quil flow. For design pUlpOses a ratio of <br />radius to width of 3 or greater is suggested for lranquil <br />flow. <br /> <br />(2) Rapid flow. Large waves are generated by r:lpid <br />now in simple curves. Therefore a much smaller rate of <br />change of curvature is required than for tranquil now. A <br />1969 stUdy by USAED. Los Angeles (1972), of as-built <br />structUres shows that curves with spiral a-ansitions. with <br />or without banked inverts, have been COnslructed with <br />radii not less than <br /> <br />4V2W <br />rm1n = -W- <br /> <br />(2-34) <br /> <br />where <br /> <br />rmin = minimum radius of channel curve <br />center line <br /> <br />v = average channel velocity <br /> <br />W = charmel width at water surfnce <br /> <br />y = flow depth <br /> <br />The amount of superelevation required for spiral-banked <br />curves (b above) is given by <br /> <br />, <br />!>y - c ~ <br />gr <br /> <br />(2-35) <br /> <br />However. this study indicates that the maximum allowable <br />superelevalian compatible with Equation 2-34 is <br /> <br />2!>y = W tan 10 = O.18W <br /> <br />(2-36) <br /> <br />or <br /> <br />!>y = O. 09W <br /> <br />2-13 <br />