The document describes optimizing the design of circular cellular cofferdams. The objective is to minimize total expected cost, which includes costs for fill material, steel sheet piling, and flooding risk. Constraints consider stability from sheet pile slipping and interlock stresses. The problem is formulated as a geometric program with four design variables - dam height, cell width, cell length, and a flooding variable. Optimal variable values are determined by solving the dual problem. The approach provides important cost and design insights for cofferdam optimization.

1. 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.

2. 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 CONNECTINGCELL
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.

3. A cofferdam design optimization 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~ ($/cu. ft.),
the cost CbLy-1Ah reasonably represents the total cost of fill and as-
sociated costs of hauling, placing and disposal.
The cost of steel sheet piles is closely related to the weight of the
steel. An estimate of the total weight of the steel is obtained by multi-
plying the steel sheet pile area density p (Lbs/sq. ft.) and the total steel
surface area Ly-1ph, where P is the cycle perimeter (ft.). Again, let Cs
be a unit cost (S/lb.) so that the total steel cost including associated
costs of transportation, setting, driving and removal of the sheet piles
can be represented by CsPLY-1ph.
The flooding risk cost is less easy to document * [3; 4]. It is propor-
tional to the (random) number of floods N during the dam life and flood
duration d (h) (also random). The total expected number of flood days
is simply dE(N), where d is the average flood duration corresponding to
the mean height h, estimated from hydrological data for the given site
[4]. Using Cf as a proportionality constant ($ per flood day), the ex-
pected flood cost is represented by CfdE(N). As an approximation :[3]
let E(N) ~ t(Clh - C2)-1, in which C1 and C2 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 - C2) -1 . (1)
* However, lives and personal property are rarely at stake for cofferdams as compared to other
civil structures. The flood risk cost is realistically measured in dollars.

4. 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~Tr-sin0)(R2-¼D 2)
+ 2RD sin2 ½0, (2)
P = 7rD + 9~TrR, (3)
Y= 2R sin ½0 +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 ~ bY 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) + Cfdt(Clh - 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 of 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

5. A cofferdam designoptimization 267
[-=-
~L F=.Pw TAN G
f )
# q~'///~.71/~¢//~"////~'1/~"4
ho
T
h
Fig. 2. Forces causing and resisting slippage.
b tan a(Pw + Po)
½Pwh +½Poho
½b tan a(Tw h2 + K'Th~) >_ 1.25,
~/wh3 + ~ K'3'h~
or
C3hb -1 + C4h-2b -1 - C5h-2 <_ 1 ,
where
5 5K' hg
C3- 12tana' C4= i2~/w tanc~
K'q,h~
, C 5 -
3'w
and
h0 = 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/cosa}~
where a is the friction angle between fill and steel piling, and ~o
is the angle of internal friction.

6. 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 tmax occurs
at the junction between the main cells and the connecting arcs (fig. 3)
and is given by [6]
tmax = ½P2 Y/cos ½0 ,
where
P2 = (Tw + K'~/)" ¼(h - h0).
The computed maximum interlock stress tmax should not exceed the
allowable stress ta specified by the manufacturers of the steel sheet
piles, i.e.,
½(h) (Y sec ~ 0) (~'w + K'7)" ¼(h -h0) -< ta (lbs/in),
or
C6 Yh - C7 Y <_ 1, (7)
where
~'w + K'3' (Tw +K'T)ho
C6 = 32 ta cos ½0 ' C7 = 32 ta cos~0

7. Acofferdamdesignoptimization 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 Clh - C2 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
h0 = 0 becomes:
minimize:
Z = (CbL)bh +(2CsPL)hbY-1 +(2CsPL)h + (Cfdt)H-1 , (8)
subject to:
C3hb-1 < 1 (slipping)
C6h Y <_ 1 (interlock tension)
C{l h-l H + C11C2h-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:
(Cbz "01 2CsPzl "02 2CsPg t #03 (Cffd} ~z04
XC3#11C6#21 ( C~-1(/131+/132))
/131
~31(CllC2(/131 +/132))
/132
#32
(9)

8. 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 - ¼U0t -- 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 - ~//32 -I- ~4//01 -I- ½//03 - 1
1 m3
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 = (CsCblp)2("yw + K'~[)(8ta cos ½0) -1 ,

9. A cofferdam design optimization
C9 = [(5 CbLCfd--t)(12 C2 tan 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 optimum of Z, Z0 (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 = (CbL)bhZO 1 , C3hb -1 = 1,
la~2 = (2 CsPL ) bh Y-1Z~ 1 ,
g'~3 = (2CsPL )hZol ,
C6hY = 1,
(12)
C-{lh-IH = U31(/a31 + g~2)-1 ,
la~4 = CfdtH-1Z~ 1 CllC2 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-

10. 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
timated at $5.67 per cu. yd. A guesstimate of $8000 is used for the
daily flooding cost. For the site in question, hydrologic data yield a
mean sample height which is exceeded, on average, 5 days during the
year. The data also yield the annual flooding frequencies shown in
table 1.
Other parameters include:
3' = unit weight of submerged material = 65 lbs/cu, ft.,
= friction angle between fill and piling = 21 ° 50',
= angle of internal friction = 28 ° 50',
K' = coefficient of active earth pressure for fill = 0.29
0 = angle between tees = 60 °.
In these data, using a least square fit, one has
E(N) = (0.8 h - 33.3) -1 .
The simultaneous equations 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 equations simultaneously 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 that before the optimum cell dimensions are known one can

11. A cofferdam designoptimization 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 61.73
= $633900.
The cell design variables are next calculated from eqs. (12) using Zo
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

12. 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-

13. 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.
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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-
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[6] W.C.Teng, Foundations design (Prentice-Hall, Englewood Cliffs, N.J., 1962).
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