Mathematical Programming 3 {1972} 263-275. North-Holland Publishing Company

A COFFERDAM DESIGN OPTIMIZATION

Farrokh NEGHABAT Bell Telephone Laboratories Inc., Whippany, New Jersey, U.S.A.

and

Robert M. STARK * University of Delaware, Newark, Delaware, U.S.A.

Received 22 December 1971 Revised manuscript received 5 June 1972

The design of circular type cellular cofferdams is formulated as a nonlinear optimization model that takes explicit account of relevant economic and technologic aspects. The objective is minimization of total expected cost. The constraints arise from stability critera to protect against failures due to slipping of the sheet piles on the river side and interlock stresses at the joint. A solution achieved with geometric programming techniques, yields optimal cell sizes and design heights. The geometric programming approach provides important design insights by yielding the proportions of the total cost to be assigned to the cost components, fill material, sheet piling and flooding in an optimal design.

1. Introduction

A cofferdam is a temporary structure built to enclose an ordinarily submerged area to permit construction of a permanent structure on the site, for example, to build a bridge pier in mid stream. The specific function (keep the construction site dry) and its short life (the project duration) contrast with the versatility and long life of most structures that concern civil engineers. The discussion is limited to circular cellular type cofferdams which are frequently used for high heads and for larger projects. The cellular cofferdam is a relatively simple structure made of fill (soil) and steel sheet piling. It is constructed by driving straight web steel sheet piles into the ground to form a series of self contained cells

* Currently on leave at the Massachusetts Institute of Technology, Cambritige, Mass., U.S.A.

264 F. Neghabat, R.M. Stark

W.T. __v____

STEEL SHEET PILES -....

OVERBURDEN

b - - - - - - - ~

~ . o o • . . . • %a. . j ".. • : ." " " . ~ v ~ : ." .'- . • .-c.~.~ < % ""F,L~ ". " " . ' ~ M

o " - ; - ' . . . • • ¢~, ~"=,K, Ke ' . ' "

~/~//ROCK / / /~///~/ '

ELEVATION VIEW

I CONNECTING CELL

0

L _ _ y _ _ _ ~ ,

PLAN VIEW

Fig. 1. Circular type cellular cofferdam.

which are filled with a suitable soil. Its analysis tends to be uncompli- cated by many failure modes.

Cofferdams function in a random environment characterized by fluctuations in surrounding water levels• If the dam height is exceeded, i.e. flooding occurs, costs of repair, replacement, clean up, dewatering, etc., are incurred besides the possibility of damage to the permanent structure. Increasing the height of the structure diminishes the chance of flooding but at an increased cost of construction• Therefore, the design height is an appropriate balance between construction and flood risk costs.

The geometry of the cofferdam is a second aspect of its design. Fig. 1 illustrates two views of a circular type cellular cofferdam. A variety of cell geometries is available to enclose the desired area. All of them must satisfy certain technologic constraints for stability and slippage to be feasible.

The conventional design of a cofferdam tends to be a trial and error procedure [7]. A trial design is chosen from experience and analyzed for feasibility. In practice, the designer manipulates this feasible design to take (economic) advanatge of site conditions as they relate to the costs of needed fill and steel sheet piles. The resulting design is feasible but not necessarily (nor likely) optimal.

The analysis to follow effectively integrates economic and technologic considerations into an expected cost objective subject to technologic constraints. Geometric programming techniques are used to derive de- sign values of the height, main and connecting cell diameters and the cycle length. The authors know of no comparably comprehensive ana- lysis.

A cof ferdam design opt imizat ion 265

2. The model

The formulation presented in this paper pertains to the "circular cellular cofferdams", resting on rock and having no inside berms (see fig. 1). The concept, however, may be applied to other types of coffer- dams.

2. 1. Ob/ective function The expected cost objective function consists of three cost compo-

nents, the costs of fill, steel sheet piles and the flooding risk cost. The fill cost is proportional to the volume LY-1Ah, where A is the

cycle area (sq. ft.) (i.e. the area of a main and a connecting cell), L the total project length (ft.), Y the cycle length (ft.) and h the dam height (ft. from river bottom). Using a proportionality constant C~ ($/cu. f t .) , the cost CbLy-1Ah reasonably represents the total cost of fill and as- sociated costs of hauling, placing and disposal.

The cost of steel sheet piles is closely related to the weight of the steel. An estimate of the total weight of the steel is obtained by multi- plying the steel sheet pile area density p (Lbs/sq. ft.) and the total steel surface area Ly-1ph, where P is the cycle perimeter (ft.). Again, let C s be a unit cost (S/lb.) so that the total steel cost including associated costs of transportation, setting, driving and removal of the sheet piles can be represented by CsPL Y-1ph.

The flooding risk cost is less easy to document * [3; 4]. It is propor- tional to the (random) number of floods N during the dam life and flood duration d (h) (also random). The total expected number of flood days is simply dE(N), where d is the average flood duration corresponding to the mean height h, estimated from hydrological data for the given site [4]. Using Cf as a proportionality constant ($ per flood day), the ex- pected flood cost is represented by CfdE(N). As an approximation :[3] let E(N) ~ t(C lh - C 2)-1, in which C 1 and C 2 are nonnegative con- stants and t is the design life in years.

The total expected cost of the cofferdam to be minimized is

Z = CIIL(y-1Ah) + CsPL(y-1ph) + Cf-dt(Clh - C 2 ) - 1 . (1)

* However, lives and personal proper ty are rarely at s take for cofferdams as compared to other civil s tructures. The f lood risk cost is realistically measured in dollars.

266 F. Neghabat, R. W. Stark

From elementary trigonometry, the geometric design variables A, P and Y can be expressed (eqs. 2 -4) in terms of the connecting cell radius R, the main cell diameter D, and the angle formed by connecting tees, 0 [3]:

A=~TrD2+ ( l ~ T r - s i n 0 ) ( R 2 - ¼ D 2)

+ 2RD sin 2 ½ 0 , (2)

P = 7rD + 9~TrR, (3)

Y= 2R sin ½ 0 +D cos½0. (4)

For convenience, designers work with a rectangular section of length Y and average width b (ft.) chosen such that its area closely approximates that of an actual cell. Therefore A ~ b Y and P ~ 2(b + Y). Substituting, the first two terms of the objective function become CbLbh and 2CsoLh(b Y -1 + 1), respectively.

The resultant objective function is written:

minimize,:

Z = CbLbh + 2CsPLh(bY -1 + 1) + Cfd t (C lh - C2 )-1 . (5)

2.2. Constraints The cofferdam design must satisfy constraints to preclude failures

[6]. Slipping of sheet piles on the river side and interlock stresses at the joints are foremost failure modes. Other relevant failure modes are sliding along the base and failure at the center line of the fill due to excessive vertical shear. Here, only slipping and interlock stress con- straints are formulated. More comprehensive considerations of techno- logic constraints appear in [3] and [6].

2.2.1. Slipping o f the sheet piles on the river side. The safety factor for slipping is the ratio between the resisting and failure causing mo- ments (fig. 2) [6, 8]. Taking moments about toe T yields

A cofferdam design optimization 267

[-=- ~L F = . P w TAN G

f )

# q~'///~.71/~¢//~"////~'1/~"4

ho T

h

Fig. 2. Forces causing and resisting slippage.

b tan a(Pw + Po)

½Pw h +½Poho

½b tan a(Tw h2 + K'Th~) >_ 1.25,

~/w h3 + ~ K'3'h~

o r

C3hb -1 + C4h-2b -1 - C5 h-2 <_ 1 ,

where

5 5K' hg C 3 - 1 2 t a n a ' C4= i2~/w tanc~

K' q,h~ , C 5 -

3'w

and

h 0 = overburden height, Po = overburden pressure, Pw = water pressure, 3, = unit weight of submerged fill, 7w = unit weight of water, K' = Coulomb's coefficient, given by

(6)

K'= [1 cos~o ]2 + {sin (¢ + a) sin ~o/cos a}~

where a is the friction angle between fill and steel piling, and ~o is the angle of internal friction.

268 F. Neghabat, R.M. Stark

l CONNECTING i CELL

. . . . i I

Fig. 3. Maximum interlock tension.

2.2.2. In terlock stresses at the join ts. The maximum tension tma x occu r s

at the junct ion between the main cells and the connecting arcs (fig. 3) and is given by [6]

tmax = ½ P2 Y/cos ½ 0 ,

where

P2 = (Tw + K'~/)" ¼ (h - h 0) .

The computed maximum interlock stress tma x should not exceed the allowable stress t a specified by the manufacturers of the steel sheet piles, i.e.,

½ ( h ) (Y sec ~ 0) (~'w + K'7)" ¼(h - h 0 ) - < t a ( lbs/in),

o r

C6 Yh - C7 Y <_ 1, (7)

where

~'w + K'3' (Tw +K'T)ho C6 = 32 t a cos ½ 0 ' C7 = 32 t a cos~0

A cofferdam design optimization 269

3. Optimization

The mathematical model described above may be solved using geo- metric programming technique [10]. While the second term of the ob- jective function of eq. (5) can be distributed to achieve posynomial form, the last term cannot. A common technique replaces C lh - C 2 by another variable, say H, and introduces the constraint

Cll h-I H + CllC2 h-1 <_ 1.

The effect of shallow overburden (a few feet, say) upon the design of the cells is usually neglected [7]. The non-linear model for the case h 0 = 0 becomes:

minimize:

Z = (CbL)bh + (2CsPL)hbY -1 + (2CsPL)h + (Cfdt)H -1 , (8)

subject to:

C 3 hb-1 < 1 (slipping)

C 6 h Y <_ 1 (interlock tension)

C{ l h-l H + C11 C2 h-1 <- 1 (change of variable)

h,b, Y , H > O.

This geometric programming primal is characterized by eight terms and four variables and, hence, three degrees of difficulty. Its dual is:

maximize:

(C bz "01 2CsP z l "02 2CsP g t #03 (Cffd} ~z04

X C3#11C6 #21 ( C~-1(/131 +/132)) /131

~31 (CllC2(/131 +/132)) /132

#32

(9)

270 F. Negnabat, R.M. Stark

subject to:

/201 +//02 +//03 +/204 = 1 (normality condition)

//01 +//02 --//11 = 0 (orthogonality condition for b)

/201 +//02 +//03 +//11 +//21 --//31 --//32 = 0 (id. for h)

-//o2 +//21 = 0 (id. for Y)

--//04 + P31 = 0 (id. for H)

all//'s > 0 .

The dual variables can be expressed in terms of//01,//03 and g32 by

//02 =//21 = ¼ + ¼/232 -- ¼//01 -- 22-//03 ,

1 //04 =//31 = ~4 - ¼ U0t -- g//32 -- ½//03 , (1 0)

/211 =¼ +¼//32 +¼//01 --~/ /03 •

The optimizing values of the dual variables //~1, g~3 and //~2 are ob- tained by (a) substituting eqs. (10) into the dual objective eq. (9), (b) setting partial derivatives of In d(//m,//o3,//32) with respect to each dual variable to zero, and (c) solving the resulting three equations (1 1) simultaneously:

C8//012//203 - ~ / / 3 2 -I- ~ 4 / / 0 1 -I- ½ / / 0 3 - 1

1 m 3 C9//03 (/232//201 )½ + ¼//01 + ¼//32 + 2//03 - ~c , (11)

C10/201//32///03 + ¼/201 - ¼//32 +1//03 = :} ,

where

C 8 = ( C s C b l p ) 2 ( " y w + K'~[)(8ta cos ½0) -1 ,

A cofferdam design optimization

C 9 = [(5 CbLCfd--t)(12 C 2 t an a) -1 ] ½(2CsPL) -1 ,

C10 = (24C 1CsP tan a)(5C2Cb) "1

271

The remaining dual variables are obtained from eqs. (10) and using /a~l, g~a and /1~2. The corresponding opt imum of Z, Z 0 (max d(/L) = min Z), which is global in view of the posynomial objective, is used to relate the dual variables to the design variables, i.e.

bt~l = ( C b L ) b h Z O 1 , C 3hb -1 = 1,

la~2 = ( 2 CsPL ) bh Y - 1 Z ~ 1 ,

g'~3 = ( 2CsPL )hZol ,

C6hY = 1, (12)

C-{ lh- IH = U31 (/a31 + g~2) -1 ,

la~4 = Cfd tH-1Z~ 1 Cl lC2 h-1 = "* Qt* + ~u~2)-1 , t~32 ~, 31 •

This is an important result for the design of the circular type cellular cofferdam under the assumptions discussed earlier. In a simple manner, the design can be explored for sensitivity to various parameters.

For deep overburden, the case h 0 = 0 may require modification. One procedure modifies the effect of overburden through inequalities (6) and (7) and a solution by trial. Alternatively, the original model can be treated but the added degrees of difficulty usually require the use of geometric programming computational codes [ 1 ].

4. Numerical example

Consider the design of a circular type cofferdam resting on rock foundation with no overburden and berm. The cofferdam will be 800 ft. long and has an anticipated life of one year. It is to be constructed of 15 inch MP 101 sections of steel sheet piling at 28 lbs/sq, ft. area den- sity with an allowable interlock tension of 8000 lbs/in. The per ton cost of purchasing, setting and driving the sheet piles into the bedrock is estimated at $163. This unit cost includes pile removal expenses and salvage value upon completion of the project. The cost of fill material, including its hauling, placing and finally its removal and disposal, is es-

272

River

h (ft) E(N)

F. Neghabat, R.M. Stark

Table 1 (dam) height versus expected number of floods

40 4 42 2 44 1 46 0.5 48 0.25 50 0.15 52 0.10

t imated at $5.67 per cu. yd. A guesst imate of $8000 is used for the daily f looding cost. For the site in question, hydrologic da ta yield a mean sample height which is exceeded, on average, 5 days during the year. The data also yield the annual f looding frequencies shown in table 1.

Other parameters include: 3' = uni t weight of submerged material = 65 lbs/cu, ft.,

= fr ict ion angle be tween fill and piling = 21 ° 50', = angle of internal fr ict ion = 28 ° 50',

K' = coeff icient o f active earth pressure for fill = 0.29 0 = angle be tween tees = 60 °.

In these data, using a least square fit, one has

E(N) = (0.8 h - 33.3) -1 .

The s imultaneous equat ions for/201,/203 and t232 are

1 0.159/2012//203 -- ¼/232 + ~/201 +g/203 --¼ = 0 ,

0.126/203 (/232//201 )~ + ¼/201 + ¼/232 + ½/203 -- ¼ = 0 ,

0.501/201/232//203 +¼/201 --43-/232 +½/203 --43- = 0 .

Solving these equat ions s imul taneously yields P01 = 0.52, Po3 = 0.25 and /232 = 1.73. The remaining dual variables using eqs. (10) are/2o2 = /221 = 0.17,/2o4 =/231 = 0.06 and/211 = 0.68.

Note tha t before the o p t i m u m cell dimensions are k n o w n one can

A cofferdam design optimization 273

assert that in an optimal design 52% of the total cost is in fill material and 42%, /%2 +/%3, in the steel sheet piles. Expected flooding costs account for only 6% of the total.

The minimum cost is

[(3.67/27)(800)] 0.52 (2 X 163x 28x 800 )o.17 Z ° = d ° = k ~ : 0.17

oo) ( ooo t × 0.25 0.06

X (1.0425) 0.68 (0.00035) 0"17

0.06

X (1"25X 1"79)°'°6 (41"63X 1"79) 1 " 7 3 0 . 0 6 1.73

= $633900.

The cell design variables are next calculated from eqs. (12) using Z o and optimal weights/z*~ The optimum cofferdam height h* is derived either from the third term in the objective or the second term of the third constraint,

whence

3651.2 h = 0.25 × 633900,

h* = 43.4 ft.

The width b is calculated from the first term of the objective,

whence

168b X 43.4 = 0.52 X 633900,

b* = 45.2 ft.

The second term of the objective function gives

274

whence

F.. Neghabat, R.M. Stark

3651.2 × 45.2 X 43.4 y-1 = 0.17 × 633900 ,

Y* = 66 ft.

Using the geometric relations (2)-(4) ,

D = 57.5 ft., R = 16.2 ft.

5. Conclusions

Structural engineers represent one of the better developed technol- ogies and they oversee an enormous investment. Only very infrequently are infeasible structural designs encountered. Yet, optimal designs are of comparable scarcity. A growing literature, dating from about the mid- sixties, attests to a recognition of the potentialities for optimization and reliability in structural analysis and designs.

Important research results have emerged and have been implemented in structural designs. However, there seem to be three main impedi- ments to more substantial progress. First, the failure modes of struc- tures can be difficult to define. Second, familiar structures have long lives and are often more versatile than originally imagined. Third, the complexity of structural analyses limits the generality with which opti- mization techniques can be utilized.

The cofferdam has important advantages as a seminal candidate for structural design optimization research. Its failure modes are distinct, its life is transient and its failure rarely results in loss of life, and the appropriate structural analyses are tractable. The preceding analysis of a circular type cellular cofferdam illustrates these observations.

The expected cost objective function and the associated constraints take explicit account of relevant economic and technologic considera- tions. The geometric programming formulation results in three simul- taneous equations whose solution is related to the design height and design geometry. Geometric programming is well known for the insights it can provide to the optimal allocation of resources among the com- ponent costs. This information provides valuable short cuts, to expe- rience and reduced calculation, for an improved design. The procedures developed here give promise for relieving much of the tedium (and ex-

A cofferdam design optimization 275

pense) of conventional trial and error besides providing an improved design.

Sensitivity analyses of parametric information is feasible and, of course, a useful source of insights. This appears to be particularly useful in this instance because cost information appears not to be well deve- loped for cofferdam design. In part, this lack of cost information reflects the transient character of the structure, the geographical variation and even the resources of the contractor. The development of design proce- dures such as the one here may stimulate better cost assessment and documentation. It is a step toward greater standardization of cofferdam design, and the core ideas appear applicable to the design of other flood control structures.

Acknowledgements

Financial support during the period of this research was provided by an NSF research grant and by the Civil Engineering Department of the University of Delaware. The authors wish to express their appreciation to Bell Telephone Laboratories and the referees for their assistance in the final preparation of this work.

References

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[2] R.J. Duffin, E.L. Peterson and C. Zener, Geometric programming (Wiley, New York, 1967).

[3] F. Neghabat, "Optimization in cofferdam design," Ph.D. dissertation, University of Delaware, June 1970.

[4] F. Neghabat and R.M. Stark, "Optimum cofferdam height," Preprint ASCE National Meeting, July 1970.

[5] R.M. Stark and R.L. Nichols, Mathematical foundations for design, civil,engineering sys- tems (McGraw-Hill, New York, 1972).

[6] W.C. Teng, Foundations design (Prentice-Hall, Englewood Cliffs, N.J., 1962). [7] Tennessee Valley Authority, Steel sheet piling cellular cofferdams on rock, Technical

Monograph No. 75, VoL 1 (1966). [8] K. Terzaghi, "Stability and stiffness of cellular cofferdam," Transactions o f ASCE 110

(1945). [9] L.W. White and E.A. Prentis, Cofferdams, second edition (Columbia University Press,

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