WIDTH AND DEPTH-CONSTRAINED BEST ...

WIDTH AND DEPTH-CONSTRAINED BEST TRAPEZOIDAL SECTION By David C. Froehlich,~ Member, ASCE INTRODUCTION

Open-channel design is often based on the assumption of uniform flow and normal depth. If a channel is to be lined with a nonerodible material the dimensions of the channel can be decided on the basis of hydraulic efficiency alone without regard to excessive velocities. From a hydraulic viewpoint, the best open-channel cross section might be one that requires the least cross-section area for a given flow rate. Computing flow rate using Manning's equation as Q = -~' A 5,3e -2,3 So1~2

n

(1)

where Q = flow rate; r = a constant (100 for SI units and 1.486 for U.S. Customary units); n = Manning's roughness coefficient; A = cross-section area; P = cross-section wetted perimeter; and So = longitudinal channelbed slope, it is apparent that if cross-section area is minimized for a given flow rate then the wetted perimeter will be minimized as well. Hence, a cross section of semicircular shape, which has the least wetted perimeter for a given area, will provide the greatest hydraulic efficiency. However, in practice easily constructed geometric shapes such as rectangles or trapezoids are most often built. While there could be economic reasons for not using cross sections of the greatest hydraulic efficiency, the dimensions of such sections might need to be known and adhered to as closely as conditions allow. Dimensions of the best hydraulic section of a specified geometric Shape are derived by King et al. (1949), Chow (1959), and in many fluid mechanics texts [see, for example, Fox and McDonald (1992)]. However, none of these derivations considers the possibility of constraints on the top width or depth of the channel cross section that site conditions will occasionally impose. The constrained minimization problem of finding the best hydraulic section is solved here using the Lagrange multiplier method with dimensionless forms of the objective and constraint functions. For the top-width and depth-constrained cases the answer requires solution of nonlinear equations. However, the nonlinear equations can be approximated closely by more easily solved expressions enabling a direct yet accurate solution. BEST TRAPEZOIDAL SECTION

The cross-section area A and wetted perimeter P of a trapezoidal channel as shown in Fig. 1 are computed as follows: 1Asst. Prof., Univ. of Kentucky, Dept. of Civ. Engrg., 212 Anderson Hall, Lexington, KY 40506-0046. Note. Discussion open until January 1, 1995. To extend the closing date one month, a written request must be filed with the ASCE Manager of Journals. The manuscript for this technical note was submitted for review and possible publication on March 12, 1993. This technical note is part of the Journal of Irrigation and Drainage Engineering, Vol. 120, No. 4, July/August, 1994. @ASCE, ISSN 0733-9437/94/00040828/$2.00 + $.25 per page. Technical note No. 5770. 828

_1 -I

W

4

~II~\\I I I F

q

b

FIG. 1. Dimensions of Cross Section of Trapezoidal Shape

A = (b + zh)h

(2)

P = b + 2hV'l + z 2

(3)

and

where b = bottom width; h = flow depth measured vertically from the bottom of the channel; and z = side-slope ratio of horizontal to vertical (z = 0 denotes vertical sides). With Q, n, and So given for a channel, determining the dimensions of the best trapezoidal cross section can be stated as a constrained optimization problem with a nonlinear objective function as follows: Minimize (b + zh)h

(4a)

[(b + zh)h] 5/3 = 0 (b + 2hX/1 + z2) 2/3

(4b)

subject to f~ _ and b -> 0

(4c)

z -> 0

(4d)

where

Qn f~ --- qb--~o

(4e)

is called the section factor for uniform-flow computation [Chow (1959); page 128], and (4c and d) = bound constraints on b and z, respectively. The problem is made dimensionless by defining relative bottom width and relative depth, respectively, as follows: b b, -= f~3/8

(5a)

h h , -= ~,~3/8

(5b)

Substituting for b and h in ( 4 a - d ) yields the following dimensionless problem: 829

Minimize (b, + zh,)h,

(6a)

subject to [(b, + zh,)h,] 5/3 1 -

(b, + 2h,~/1

+ 2'2) 2/3

= 0

(6b)

and b, -> 0

(6c)

z >- 0

(6d)

The dimensionless objective function [(6a)] and dimensionless constraint function [(6b)] are combined to form an augmented objective function (the Lagrangian function) as follows: [(b, +

L(b,, z, h,, k) = (b, + z h , ) h , + k

zh,)h,]5/3 }

1 - (b, + 2 h , V ' l + z2) 2/3 (7)

where h = a Lagrange multiplier. At an extremum, the following conditions need to be satisfied from principles of differential calculus [Sokolnikoff and Redheffer (1966); page 342]:

OL Ob-~, = 0;

OL = O; OL OL 0-7 0h, = 0; - ~ = 0

(8a,b,c,d)

The four resulting equations are solved simultaneously for the following optimal values of z, h,, and b, for a channel of trapezoidal shape without top-width or depth constraints: 1

z = V~

(9a)

h, = [(1 - z 2 + 2zX/1 + z2)z3/2] TM = 0.968

(9b)

1118

9c,

w, = b, + 2zh, = 2.235

(9d)

The optimal relative top width is then

which is twice b,, giving the well-known solution that the best (unconstrained) trapezoidal section is one-half of a regular hexagon. Top-Width-Constrained Case

If the relative top width of a trapezoidal section is limited by site conditions to a value ~ , less than 2.235, the optimal values of z, b,, and h, will not equal those given in (9a-d). The optimization problem is the same as given by (6a-d) except for the addition of the constraint 830

v~ - (b, + 2zh,) = 0

(10)

The dimensionless Lagrangian function for this problem becomes

[ L ( b . , z , h . , k ~ , h 2 ) = (b. + z h . ) h . + hi

[(b* +zh*)h*) 5t3 ] 1 - (b. + 2 h . X / 1 + z2) 2/3

-~- ~.2{1~, -- ( b , --t- 2 z h , ) }

(11)

where hi and hi = Lagrangian multipliers associated with the two constraint functions, ~, -= vP/l)3/8; and ff = specified top width. The following conditions need to be satisfied at an extreme value:

OL = O; --OL= O; __OL= O; __OL= O; __OL= 0 Ob. Oz Oh. Ok1 Oh 2

(12a,b,c,d,e)

Solving (12a-e) yields the following expressions for the optimal values of z, h,, and b, for the case of a top-width-constrained trapezoidal channel: 1

1 + z~]

[(

z~ 1-1

]5,3

:~z~)i-~J

= ~*

(13a)

)

and

b, = vP, - 2zh,

(13c)

where = V'I + z 2 - z

(13d)

Eq. (13a) is solved first for z, then h, and b, are computed directly from (13b and c). The relations between z, h,, b,, and ~ , are presented graphically in Fig. 2. Notice that, for rP, -< 1.316, the side-slope ratio z is constrained by its lower bound of zero. Solution of (13a) for z requires iteration, which might prevent a speedy calculation. However, the relation between z and vP, given by (13a) is closely approximated by the following expression (determined by regression analysis): z = -0.612 + 0.367~, + 0.0740v~2,

(14)

for 1.316