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e . . e . e graphs and tables have been published to facilitate compu- tation of unifonn flow. Brater and King (1976) have specially prepared tables for trapezoidal channels based on the Manning equation. HDC 610-1 through 610-4/1-1 give graphs that afford rapid solution for the nonnal depth in trapewid channels. Nonunifonn or varied flow in prismatic channels can be solved rnpidly by use of the varied flow function. (It should be noted that different authors have used the tenns "nonunifonn" flow and "var- ied" flow to mean the same thing; "varied flow" is used in this manual.) Varied flow in nonprismatic channels. such as those with a gradually contracting or a grndually ex- panding cross section. is usually handled by "step meth- ods." It should be noted that short, rapidly contracting or expanding cross sections are treated in this manual as transitions. (a) Prismatic channels. A prismatic channel is char- acterized by unvarying cross section. constant bottom slope. and relatively straight alignment. There are three general methods of detennining flow profIles in this type of channel: direct integration. direct step. and standard step. as discussed in Chow (1959. pp 252-268). The direct integration and direct step methods apply exclu- sively to prismatic channels. whereas the standard step method applies not only to prismatic channels but is the only method to be applied to nonprismatic channels. The direct integration method (with certain restrictions as to the constancy of hydraulic exponents) solves the varied flow equation to detennine the length of reach between successive depths. Use is made of varied-flow.function tables to reduce the amount of computations. This method is not nonnally employed unless sufficient profIles and length of channel are involved to warrant the amount of precomputational preparation. The direct step method detennines the length of reach between successive depths by solution of the energy and friction equations written for end sections of the reach. The standard step method is discussed in (b) below. (b) Nonprismatic channels. When the cros~ section. alignment, and/or bottom slope changes along the channel. the standard step method (Chow 1959. p 265) is applied. This method detennines the water -surface elevation (depth) at the reach extremity by successive approxima- tions. Trial water .surface elevations are assumed until an elevation is found that satisfies the energy and friction equations written for the end sections of the reach. Cross sections for this method should, in general. be selected so that velocities are increasing or decreasing continuously throughout the reach. EM 1110-2-1409 contains further infonnation on this method. Plate 8 shows a sample computation for a gradually contracting trapewidal EM 1110-2-1601 1 Jul 91 channel where both bottom width and side slope vary. Successive approximations of water-surface elevations are made until a balance of energy is obtained. Friction losses "r are based on the Manning equation. Sf = n2V2 2.21R4/.l (2-1 and 2-2 bis) v2 = C2R For the sample computation a mild slope upstream and steep slope downstream of sta 682+40 have been assumed. Critical depth would occur in the vicinity of sta 682+40 and has been assumed as the starting condi- tion. Initially, column 21 has the same value as column 10. The computations proceed downstream as the flow is rapid. The length of reach is chosen such that the change in velocity between the ends of the reach is less than 10 percent. The energy equation is balanced when column 21 checks column 10 for the trial water surface of column 5. Plate 9 repeats the computation. substituting k = 0.002 ft for n = 0.014. For rough channel conditions C = 32.6 10gtO C2~2R) (2-6 bis) 2-3. Flow Through Bridges Bridge piers located in channels result in energy losses in the flow and create disturbances at the bridge section and in the channel sections immediately upstream and down- stream. As bridge pier losses materially affect water- surface elevations in the vicinity of the bridge. their careful detennination is important. Submergence of bridge members is not desirable. a. Abutment losses. Bridge abutments should not extend into the flow area in rapid-flow channels. In tranquil-flow channels they should be so designed that the flow depth between abutments or between the abutment and an intennediate pier is greater than critical depth. The Bureau of Public Roads (BPR) (Bradley 1978) has published design charts for computing backwater for various abutment geometries and degrees of contraction. The design procedure and charts developed by BPR are recommended for use in channel designs involving bridge abutments. For preliminary designs. a step backwater computation using abrupt expansion and contraction head losses of 1.0 and 0.5, respectively, times the change in 2-5