INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, VOL. 13, 599-610 (1989)

SYSTEM ANALYSIS I N CALCULATION OF CANTILEVER RETAINING WALLS

EUGENIUSZ DEMBICKI A N D TRAN CHI

Technical Universily of Gdansk, Wydzial Hydrovchniki, ul. Majakowskiego 1 I , 80 952 Gdansk- Wrzeszcz, Poland

SUMMARY

The paper is concerned with the solution and determination of the optimum shape of a retaining wall modelled by the co-ordinates of a set of points which belongs to a certain permissible space.

An optimum design method is formulated as a non-linear multiobjective optimization problem. The following two objective functions, which are mutually conflicting and non-commensurable with each other, are considered: (a) to minimize the total weight of the wall, (b) to maximize the structure stability.

A set of Pareto optimum solutions is derived numerically by adopting a logical algorithm, and a Pereto optimum solution is obtained by investigating the trade-off relationships among design objectives.

INTRODUCTION

Figure 1 shows a model of the retaining wall structure. In recent years many studies on the geometrical relationships of the parameters (a, p, d, b, B, t ) of the retaining wall have been carried out. Studies provided by many authors'-4 indicate that the changes in these parameters of a retaining wall (Figure 1) considerably affect its stability.

Parameter studies conducted to examine the relationships among them were carried out for n parameters separately (or for each double parameter (b/B)). Thus, interactions with each other and among all the parameters have been ignored. On the other hand, in the above works, a single objective function such as the total weight or stability, is reviewed while other evaluative factors are all considered as constraint functions. In such a methodology, the complicated relationships among many evaluative factors are difficult to assess, and optimum design solutions derived from a comprehensive view of the wall cannot be obtained. To overcome these difficulties, we will use a simulation system based on direct Monte Carlo simulation using the boundary representation code to consider the change of the retaining wall geometry. Herein, all parameters of the wall are considered together as a single geometric unit, and the different objectives mentioned above, which conflict with one another, are considered. It is possible to develop a sensitivity analysis formulation for the rates of the relationship between the wall's parameters with respect to the Optimization design. A multi-objective optimal approach will be proposed, in which a method for obtaining a Pareto-optimal solution set in order to evaluate the conflicting objectives in an analysis of a retaining wall is used. The Pareto-optimal solution is taken from the Pareto-optimal set based on the explained trade-off relationships between conflicting evaluative characteristics of the wall, through the minimax principle. The analysis in this study provides a more complete mathematical description and a better understanding of the geometrical behaviour of such a structure than solutions which were used previously. A numerical example for this analysis is shown separately.

0363-9061/89/060599-12$06.00 0 1989 by John Wiley & Sons, Ltd.

Received 5 January 1987 Revised 21 January 1988

600 E. DEMBICKI AND T. CHI

Figure 1. Design variables of a retaining wall

MATHEMATICAL DESCRIRTION OF THE GEOMETRICAL CHARACTERISTICS OF THE CONSTRUCTION

Geometry of the retaining wall and boundary representation

Any geometry of the retaining wall exists within an embedding space with dimensionality equal to the minimum number of co-ordinates required to describe each and every point of construction of the wall uniquely. Assume the wall can be represented by the co-ordinates of a set of ( n = 4 ) specific points, in which their horizontal co-ordinates are described by the ordered set

H=(x1, x3, x5, x7) (1)

V=(x2, x4, x 6 , x8) (2)

and the vertical co-ordinates are described by

The position of any point is defined by the pair of ordered numbers ( x i , x i + in H 0 V, such that xirxi+ where 0 is called the Cartesian product

and it is a relation ( r )

H@ V = { ( X ~ , X ~ + ~ ) ~ X ~ E H , X ~ + ~ E V } , i = l , 3, 5 , . . . (3) Figure 2 shows the orderly sequence of pairs of points imparted.

From the geometrical point of view, all co-ordinates are treated equally, even if they have different physical interpretations. We have a model of the retaining wall, which is represented by its boundary line. The construction operation in the boundary representation of the walls ensures that the boundary is well formed in the sense that it is closed and orientable. The method for creating and manipulating the geometry in the boundary representation of the wall includes sweeping, which defines the boundary line of the wall by moving a point along a path in space, and ‘tweaking’ which performs local operations on the geometry but leaves the topology of the model intact. The geometry of the wall is completely specified by a co-ordinate vector of n independent components, and is obtained by formulation

G = X c H @ V (4)

SYSTEM ANALYSIS 60 1

Figure 2. Simulation of the structure of a retaining wall: (a) symbols; (b) determination of co-ordinates of characteristic points

where

x=(xl , x 2 7 x37 x 4 7 * * ,x8)

and 0 represents the Cartesian product. The topology of the wall is obtained from the following formulation:

T={XIXEE1, E , c R " )

where El is defined by the following restraining conditions:

(5 )

(7)

g l ( x ) = x i > O , i= 1,2,. . . , 8, (x4=x,=O)

9 2 (4 = x1> a1

94(x)=xg-x32a,

g5(x)= x7 - x5 3 a3

g,(x) = xz - H , z 0

g3(x)=x3-x1 ? O

g7(x)=Xg-x220

Consider the region E , as the permissible space in which the evolution of the retaining structure takes place. This space is defined by the following restraining conditions:

Structure constraints E ,

g*(x)=b1 G X l Gbz

g9(x)=x3Gxl +x, tan 10"

602 E. DEMBICKI AND T. CHI

Stability constraints E,

where R" is n-dimensional Euclidean space, Hp is the horizontal component of the load resultant acting on the foundation of the wall (N, is the vertical component), EMu is the sum of the anticlockwise moments, ZMn is the sum of the clockwise moments. Q is the vertical component of the bearing capacity of the subsoil under the retaining wall, R, is the vertical component of the load resultant, 6 is the inclination of the load resultant with respect to the normal to the foundation plane, H o is the surcharge of set height, MI is the resisting moment of the soil and wall mass as a whole and M , is the overturning moment of the soil and wall mass.

Design objectives

As mentioned in the introduction, it is fundamentally desirable to investigate multiple design objectives in the analysis of the geometry of the retaining wall studied in this paper. Two elements are considered as design objectives:

1. To minimize the total weight of the retaining wall

fl = { (x 5. x, - x 3 . x ,)( 1 /cos f l ) + (x5. x7 - x 3 . x7) l/cos a ) yc +minimum (10)

where f l= tan-' ((xj - xl)/xz)

and y c is the mass density of the wall. a = tan-' ((xg -x2)/(x7 -xl))

2. To maximize the stability of the wall

f , = Q/R, cos 6 =f,(xl, x2 , . . . , x,)-+maximum (1 1)

However the two objectives minimizing the total weight of the wallf, and maximizing stabilityf,, conflict with each other. Therefore the minimization offi and maximization offi cannot be obtained simultaneously, as shown in Figure 3. In this case optimization requires a Pareto optimal set showing a trade-off relationship between the two objectives.

As maxf(x) = -min( -f(x)), the set of all objective functions may be written as

F = If, (x),f,(x)) -+minimum (12) From the adopted formulation of the objective functions (lo), (1 1) and restraining conditions (7)-(9) of the problem, the geometrical optimization of the wall becomes

minimum { F 1 F = (fi ,L)J XEE

E = { x l x ~ El n E , n E3}

SYSTEM ANALYSIS 603

F

0.019 0.010 0.017-

A - I \

_ - - _ _ _ - -

44 65 75 90 115 19 29 ITERATION

Figure 3. Competitive situation between two objective functions

In this model the overall shape of the retaining wall is determined by an assemblage of points whose co-ordinates are required. It has dictated the evolution of the geometry of wall, regarding which it should be mentioned that the topological constraints (7) ensure the capability of self- reproduction, that is the capability of the construction to produce copies of these, and the conditions (8), (9) ensure the possibility of changing certain structural properties of the con- struction (geometry) with simultaneous preservation of the self-reproduction capacity (topology). The optimization problem proposed previously is very complicated because both the constraint relationship (9) and the objective functions (lo), (11) are non-linear with respect to unknown variables, and vector-valued objective functions (1 3) should be minimized. To overcome this difficulty, the following three process are introduced:

(a) Determining the objective space Y

Y = { f i ( x ) I x ~ E } , i = l , 2 (15)

f p = inf {fi’(x)lx E E , i = 1,2} (16)

P c Y (17)

and the minimum of the objective functions for each of the objectives separately:

(b) Determining the Pareto-optimal solution set P

(c) Seeking the solution optimum in the sense of compromises (x+, F ( x + ) ) or the preference solution:

SOLUTION OF MULTI-OBJECTIVE OPTIMIZATION PROBLEM

Determining the objective space Y and minimum of the objective functions separately f y , f 4 Notice that mapping (see Figure 4) takes a unit square uz into two-dimensional space via a

604 E. DEMBICKI AND T. CHI

parametric mapping, and its changes allow the description of geometric evolution as a natural extension.

We may introduce the following transformable function: . .

h = x ' E x : + ( X ~ - X ; ) J / i , i = l , 2 , . . . , 8 (19) where x8 and x i are the maximum and minimum limits of the variable x i and J/' is a random number characterized by a uniform distribution in the interval (0, l), or evolutionary factor.

Because each point x E E is defined by an appropriate point uje u8 through the function h, therefore

h : u - + E (20)

F : E + Y , F=(f,,f,) (21)

F 0 h : u - r Y (22)

From the other side, we have

From (20) and (21) it follows that

Geometric computation of the retaining wall is, in fact, the derivation of its explicit geometric representations from complete but implicit geometric input data, Thus the first stage towards automation requires the preparation of a subprogramme to generate parameters for the wall in terms of the chosen design variables. The simulation of the Monte Carlo method is used for simultaneous determination of the objective space Y and evaluation of the extreme points of all of thefi', i = 1,2. Figure 5 shows the general procedure for the geometrical evolution of a retaining wall.

Determining the Pareto-optimal solution set P c Y

In our paper the concept of a semi-order relation is generally used to define the optimality. It is a fundamental approach that is based on the concept of Pareto optimality. A Pareto-optimal solution is qualitatively defined as one where any improvement of one objective function can be achieved only by permitting the deterioration of another.

A Pareto-optimal solution. Assume that xo EE is the solution of problem (13), (14) such that

F(xo) =minimumF(x)

where F ( x o ) = (fi ( x o ) , f 2 (x')), x o = (xy , xg , . . . , x:). There is no such other point x + E E for which

F(X+ ) >F (XO), F(x + ) # F(xO)

Yl Y

Figure 4. Mapping takes a unit square u into two-dimensional space

SYSTEM ANALYSIS 605

Figure 5. General procedure for geometrical evolution of a retaining wall

we therefore have

From this definition we can see that xo E E is a Pareto-optimal solution. There is no feasible design that can provide a simultaneous improvement for all the objectives.

The Pareto-optimal solution set. Assume that the relative importance between the two objective functions is known:

(F’ f w++ ( ( f l ( . ’ )#f l (XJ)) v (fi(x’)#fi(xj))) (25)

Thus in trade-off relationships among two objective functions, we have

It is evident from (26) and (27) that these points F(x’), F ( x j ) will not be compared. From definition (24) we have F(xi ) , F ( x j ) , which are the Pareto-optimal solutions. For convenience, we will denote the set of all Pareto-optimal solutions by P.

Formally, a point F(x’)E Y is Pareto optimal if

and

From (28) for all the points F(x’)E Y which is not a Pareto-optimal solution, we always have F ( X ~ ) E P t Y. So F ( x j ) > F ( x ’ ) and they can be expressed by

Determining the Pareto-optimal solution set. Usually, as Pareto-optimal solutions for multi- objective optimization problems consist of infinite numbers of points, in general not all of them are obtained in a practical numerical c a l ~ u l a t i o n . ~ ~ ~ As shown in Figure 5 there exist some points which are not comparable with regard to one or another design effect. In this paper we will define a Pareto-optimal solution set by eliminating all points that do not belong to it, with the following ‘refinement’ operation.

606 E. DEMBICKI AND T. CHI

Abstract treatment of the problem 1. The objective point selected is F ( x ' ) and the point compared is F ( x 2 ) : F ( x ' ) , F(x')E Y

F ( x ' ) = ( f i ( x ' ) , h ( x l ) , . * . f k ( X 1 ) ) ,

F ( X 2 ) = ( f i ( X 2 ) , f 2 ( X 2 ) . . . f k ( X 2 ) ) ,

k 2 2

k 2 2

i = 1 , 2 , . . . m x ' = ( x ' , , x : , . . . , x:), Notation: instead of ( x l , F ( x l ) ) > ( x 2 , F ( x Z ) ) we write

F ( x ' ) > F(X2)

Definition: let E and Y by preference ordered sets. A mapping F , from E , to Y is called an order homomorphism if x 1 < x z implies F ( x l ) < F ( x 2 )

2. p = ( f i ( x i ) > f i ( x J ) ) = 1, l ( J ( X ' ) > f i ( x j ) = O (30)

q = ( f i ( x ' ) > h ( x j ) ) = 1, 1 (f2(xi) > f i ( x j ) = O (31)

1 (fibi) H&4)4mi) < f i ( x j ) ) (32)

1 ( f 2 ( x i ) ~ f 2 ( x j ) ) - ( f 2 ( x i ) ( f 2 ( x j ) ) (33) Comparing the objective points, the following three cases may occur:

A {.&(x') >.&(x')}, i= 1,2,3, . . . , k is1

(34)

(35)

where the symbol '>' denote the preferred relation and is read as is 'better' or 'predominates', etc. In the first case we know for example that the point x 2 does not belong to the Pareto-optimal solution set. However, it should also be mentioned that the point x 1 cannot be regarded as belonging to the Pareto solution set, because it should be compared with other points located in the permissible space Y. Accordingly, for the second case the point x 1 does not belong to the Pareto solution set. In the third case both points x l , x 2 may possibly belong to the Pareto set (but it is not certain). It follows then that it is difficult to determine a point belonging to the Pareto set. It is easier, however, to determine points that do not belong to the Pareto set. Let N P denote a set of points that do not belong to the Pareto-optimal solution set, then

N P u P = Y (37)

(38)

From (30)-(33) and the de Morgan principle

{ 1 ( P A 4 ) 4 ( 1 P ) v (14)) the procedure is easily derived.

F ( x ' ) E N P when

~ 1 = { ( f , ( x ' ) > ( ~ ( x 2 ) ) v ( f i ( x 1 ) > f 2 ( x 2 ) 1 v . . . v ( f k ( x 1 ) > f k ( x Z ) ) 1 = O (39)

SYSTEM ANALYSIS 607

F(x2) E N P when

Therefore we have the following operation:

NP= A {F(x'):F(x')E YIL, = 0, L2 = l} x i s E

i.e.

N P = A {F(x')c Y ( V F(x j ) > F(x'), izj} X i € E x j s E

From (37) we obtain

P = Y\NP (43)

The comparison of objective functions is based on the value of relative deviation 2, (x) from the extreme solution fi :

To find solutions characterized by the minimum value of the relative deviation from the extreme solutions we assume that

Z'(X1)<Zi(X2)++fi'(X1) >J(x2) (45)

Seeking the preference solution (x', F(xf)) . For any case of practical usefulness determination of the Pareto-optimal solution set can only be an intermediate step in the decision making. At the end it is always selection of the preference solution, possibly versatile under the conditions given, which is the ultimate objective of decision making.

Calculations are carried out according to formula (44) for each objective (i = 1,2, . . . , k ) and for x=xj. A vector Z j i is obtained

zji = {zjl (x),Zjz(x), . . . 9 zjk(x)} (46)

The calculations are performed for m variants: thus a rectangular m x k matrix Z is obtained

Subsequently, the matrix q5 can be obtained from

where

$ji=IZji-maxZji(, I = 1, 2, . . . , k is1

608 E. DEMBICKI A N D T. CHI

we regard the matrix q5 as a payment table of a game for two. We select from each row the elements

max4ji=sup(4ji , I = l , 2 , 3 , . . . , k } ioI

max4= i e I

max

max

max

i e l

i o I

ioI

. . . , m yielding

42i , I = l , 2 , . . . , k

I (49)

On the basis of the multiobjective design optimization we take a vector (x+:F(x+)€P) as the solution of the problem (13), (14) when it satisfies the condition

4(x+)=minmax+ji(x), M = l , 2 , . . . , m j s M i o I

Now the problem (13), (14) can be expressed as follows: Find

x+= [ x l )

xn

which minimizes max (bji(X) , for j E M . Thus, ( i s 1 )

E={xJxE(El n E , n E , ) } (54)

A problem such as (51)-(54), is regarded as an hierarchical multi-objective decision system, in which there exist k distinct local objective functions A, i = 1,2, . . . , k and a super-objective function 4 in which the parametrized constrained optimization problems of the lower level are contained as part of the constraints of the upper level.

The lower level generates the Pareto-optimal solution set (X = { ($1 F ( X ) E P } . As local decision makers do not negotiate, competitive situations are assumed not to exist. The central decision maker in the upper level determines the best non-inferior solution (x', F ( x + ) ) based on its own objective function.

NUMERICAL EXAMPLES

Numerical data shown in Table I are used for the following numerical example. Figure 3 shows a competitiveness between the two objective functions: minimizing the total weight fi and

SYSTEM ANALYSIS 609

Table I. Data for numerical study

4 0 5 n1

a3 0 5 n3

b, 0.5 YG

b2 3 4 b3 0.45 C b4 0.5 Ho b5 0 5 4 b6 6 Yc

a2 0.45 n2

n4

1.5 1.5 3 1.5

18 kN/m3 30" 0 7

10 kPa 24 kN/m3

160

150

140

130

Figure 6. The Pareto-optimal solution set (29, 65, 76, 19, 90, 115, 44)

maximizing the stabilityf, of the retaining wall. The Pareto-optimal solution set is shown in Figure 6. Figure 7 shows a trade-off relationship between the two objectives. Numerical data presented in Table I1 are general results which were obtained by the method mentioned above. Variant 10 is characterized by simultaneously minimum material volume and maximum stability of the structure (115 iterations, Figure 7).

CONCLUSIONS

In our study on the optimization of the geometry of a retaining wall, the effect of interaction between the parametric dimensions of the wall is investigated using a systems approach where the parametric dimensions are considered together as a single unit. Such an analysis in this study provides a more complete mathematical description so that it allows the process of seeking the new geometry of a retaining wall to be automated. Two practical competing design objectives are formulated with relevant applicable results to cover the most practical application of the wall, and it could be extended to the analysis of other structures with more objective functions. It has been observed that optimization with respect to only one of the performance characteristics would eventually yield impractical results. However, consideration of the interaction between the retaining wall and its foundation with respect to relative wall displacements, wall stiffness and its reliability should be a topic of studies in the future.

610 E. DEMBICKI A N D T. CHI

1.9 2.9 4.4 65 Z6 90 115 DECISION

Figure 7. Trade-off relationship between two objectives

Table 11. Examples of general solutions

fi fi B ci x 2 x3 x4 x S x6 x7 x S

1 2 3 4 5 6 7 8 9

10

168.48 148.39 148.39 147.95 147.94 146.1 1 146.39 130-8 1 159.67 150.67

6.16 14.28 15.08 14.91 4.24 9.05

14.54 3.38

18.89 15.91

0.00 + 2.66 - 934 -2.85 - 5.00 + 9.75 - 2.48 - 040 + 9.98 - 8.94

0.00 1.5 9.05 1.5 0 1.95 0 6.55 9.05 8.12 092 8.05 1.3 0 1.75 0 5.63 8,7

10@0 2.86 8.57 1.45 0 1.90 0 4.98 8.9 8.63 1.35 8.08 095 0 1.40 0 5.54 8.7 0.00 1.32 8.74 0.56 0 1.01 0 4.92 8 7 5.96 0.56 8.21 1.96 0 2.41 0 5.17 8.6 8.24 1.29 8.02 0.94 0 1.39 0 5.47 8.6 0.00 0.56 8.05 0.50 0 0.95 0 4.06 8-0

10.00 074 8.01 2.15 0 2.60 0 6.49 9.0 10.00 2.38 8.27 3.64 0 4.09 0 5.50 8.8

1.

2.

3.

4.

5.

6.

REFERENCES

A. Danilewski, ‘Safety factor of dams and retaining walls’, J . Geotechnical Engineering Division ASCE 108. (GTI), 47-61 (1982). E. Dembicki and A. Tejchman, Wybrane Zagadnienia Fundamentowania Budowli Hydrotechnicznych, PWN, Warszawa- Poznan, 1981. E. Dembicki, W. Cichy and M. Jedrzejczyk, Zagadnienia Statecznoki i Optymalizacji Kqtowych Nabrzeiy Oporowych, Iniynieria Morska, 5/1984. G. Martial, ‘Action des terres sur les ouvrages de soutenements-stabilitt des murs poids’, 7’hPse: Docteur-ingknieur, Grenoble, 1972. K. Ito, S. Akagi and M. Ohta, ‘A multi objective optimal design of thermal distribution systems’, Journal ofMechanisrns. Transmissions, and Automation in Design, A S M E , 106, 142-147 (March), (1984). M. Peschel and C. Riedel, Polioptymalizacja, Wydawnictwo Naukowo-Techniczne, Warszawa, 1979.