OPTIMIZATION AND SENSITIVITY OF RETAINING STRUCTURES

By A§km Sarlba§l and Fuat Erbatur2

ABSTRACT: This paper is concerned with optimum design and sensitivity of retaining structures. The optimumdesign formulation in terms of a constrained nonlinear programming problem, is given for reinforced concrete­cantilever retaining walls. The objective function may be chosen as the cost or weight of the wall. The solutionis carried out by a specially prepared computer program (RETOPT). Illustrative problems are solved, and theirresults are presented and discussed. The formulation allows for a detailed sensitivity analysis to be made forselected design parameters, also depicted with numerical examples.

INTRODUCTION

This paper presents the results of part of a comprehensiveinvestigation concerned with optimum design of retainingstructures. In this regard, formulation and solution methodol­ogy is presented for reinforced concrete-cantilever retainingwalls (Fig. I). The optimum design problem is posed as aconstrained nonlinear programming problem that is solved bya specially prepared program. Two objective functions,namely, the cost and weight of the wall are considered. Sevendesign variables representing the cross-sectional dimensionsand reinforcing steel are chosen. The programming problem iscomplemented by 24 behavioral and side constraints.

Optimum design of retaining walls has been the subject ofa number of studies: Alshawi et al. (1988); Dembicki and Chi(1989); Fang et al. (1980); Keskar and Adidam (1989); Pocht­man et al. (1989); and Rhomberg and Street (1981). Thesestudies deal with various aspects of optimal design, including,optimal shapes. maximization of structural stability, minimi­zation of bending moments, and optimum location on slopinghillsides. The main features of the present study are: it is com­prehensive, assumptions are kept to a minimum, the computerprogram is used with ease, optimal design is automaticallyreached, the computation time is very short, and it is flexiblein formulation allowing for further modification and extension.

Two example problems are presented. The parameters of theexample problems are tabulated in Tables 3 and 7. The optimalsolutions include: objective function values, amount of steeland concrete, magnitude of design variables, and magnitudesof constraints. The results are given in Tables 4-6 and Tables8-10.

The solution approach readily allows to study the sensitivityof optimum design. The sensitivity of the optimal solution maybe searched with respect to a variety of design parameters,which include those related to soil properties, loading, andmaterial characteristics. The results of sensitivity analyses forheight and top thickness of the stem, surcharge load, backfillslope. internal friction angle of retained soil. and the yieldstrength of reinforcing steel are presented (see Tables II and12 and Figs. 2-7).

BEHAVIOR OF RETAINING STRUCTURES

The study of retaining structures falls in the domain of thegeneral soil-structure interaction problem. This is concerned

'Res. Assoc.. MS in Civ. Engrg.• Dept. of Civ. Engrg.• Middle EastTech. Univ., 06531 Ankara, Turkey.

2Prof.• PhD in Civ. Engrg., Dept. of Civ. Engrg., Middle East Tech.Univ.• 06531 Ankara. Turkey.

Note. Discussion open until January I, 1997. To extend the closingdate one month, a written request must be filed with the ASCE Managerof Journals. The manuscript for this paper was submitted for review andpossible publication on August 10, 1995. This paper is part of the Journalof Geotechnical Engineering, Vol. 122, No.8. August. 1996. ©ASCE,ISSN 0733-9410/96/0008-0649-0656/$4.00 + $.50 per page. Paper No.11369.

with the interaction of the retaining structure and the surround­ing soil. The primary aim of the problem is to explore theconditions supporting the backfill safely, with controlled dis­placements of the specific retaining structure and the backfillitself. The general trend followed in the design of such struc­tures is to ensure both that the structure and the soil surround­ing the structure do not fail and that the deformations that takeplace are within acceptable limits. These requirements are usu­ally met by avoiding the possibility of overturning and slidingof the structure, by controlling the stresses within allowablelimits, and by safeguarding the stability of both the structureand the soil.

REINFORCED CONCRETE-CANTILEVER RETAININGWALLS

The reinforced concrete-cantilever retaining wall is the mostcommon of the retaining structures. The wall has a simplegeometry, consisting of a base slab and a stem. It is made inthe form of an inverted T or L. The main design principles,briefly mentioned in the previous section, are also valid forreinforced concrete-cantilever retaining walls. Specific to these

H

I} tWs

:w,.1: .w2 :

IIII

X4

X

x,-X 2 -X 3

X 2 x1 - X 2

Xl

FIG. 1. Mathematical Model Used for Optimum Design of Re­Inforced Concrete Cantilever-Retaining Wall

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TABLE 1. Failure Modes

FORMULATION OF OPTIMUM DESIGN PROBLEM

Constraints

(1)

(3)

(2)

gj(X) S 0, j = I, ... , 10

pressed as functions of the design variables and correspond tothe 10 behavior constraints, defined as inequalities

TABLE 2. Lower and Upper Bounds of Design Variables

'Local market rates.

Lower bounds Upper bounds(1) (2)

X,,,,,. = OAH(l2/1l) X,_ = (0.7H)/0.9X2"". = [OAH(12111»)/3 X2_ = [(0.7H)/0.9)/3X3"". = t X3_. = (H/0.9)110X..... = [H(12111»)/12 X4_. = (H/0.9)/1OX,,,,,. = 10,000 Pm..(t - 0.01 m) X,_. = 10,000 Pm~(X3_. - 0.01 m)X..... = 10,000 Pmln(X4"". - 0.01 m) X"-. = 10,000 Pm~(X4_. - 0.01 m)X7"". =10,000 Pm..(X..... - 0.01 m) X7_. = 10,000 Pm~(X4_. - 0.01 m)

Note H = Height of stem. Pm.. = Minimum steel ratio. Pm~ = Maximumsteel ratio. m = Clear concrete cover + half of diameter of reinforcingbars.

TABLE 3. Input Parameters for Example 1

[(Xi)min/X;] - ISO, i = I, ... , 7

for minimum values and

where x =vector of design variables.The derived constraint expressions are found to be highly

nonlinear in the design variables (Sarlbll§ 1995). In addition,all the design variables have practical minimum and maximumvalues (ACI 1990; Gaylord and Gaylord 1990; and Bowles1988), which are shown in Table 2. These side constraints arealso expressed as inequality constraints

for maximum values.Thus, the total number of design constraints involved in the

optimum formulation is 24.

Objective Functions

Two objective functions, namely, the weight and cost, havebeen chosen for flexibility of use and for comparison purposes.In cost minimization the objective function is defined as

Input parameter Unit Symbol Value(1 ) (2) (3) (4)

Height of stem m H 3.0Top thickness of stem m t 0.20Yield strength of reinforcing steel MPa I, 400Compressive strength of concrete MPa j; 21Wide beam shear strength of concrete MPa ~, 0.65Concrete cover cm d, 7Maximum steel percentage - Proal( 0.016Minimum steel percentage - Pmin 0.00333Shrinkage and temporary reinforcement

percent - PST 0.002Diameter of bars cm <1>"", 1.2Surcharge load kPa q 20Backfill slope degree 13 10Internal friction angle of retained soil degree <1> 36Internal friction angle of base soil degree <1>' 0Unit weight of retained soil kN/m3

"I, 17.5Unit weight of base soil kN/m3

'Y: 18.5Unit weight of concrete kN/m3

'Yc 23.5Cohesion of base soil kPa c 125Design load factor - LF 1.7Depth of soil in front of wall m D 0.5Cost of steel- $/kg C, 0040Cost of concrete' $/m3 C, 40Factor of safety for overturning stability - N, 1.5Factor of safety against sliding - N, 1.5Factor of safety for bearing capacity - SF 3.0

Failure mode(2)

Shear at bottom of the stemMoment at bottom of stemOverturning stabilitySliding stabilityNo tension condition in foundationBearing capacityToe shearToe momentHeel shearHeel moment

g,(X)g2(X)g3(X)g,<x)g,(X)g.(X)g7(X)g,<x)g9(X)glO(X)

Inequality constraint(1 )

structures, the following failure modes should be considered(Keskar and Adidam 1989; Rhomberg and Street 1981): notension condition in the foundation, sliding stability, overturn­ing stability (about the toe), failure in bearing capacity, andfailures related to the shear and moment capacities of the threemain components of the wall, namely, the stem, toe, and heel.

The general three phases considered in the optimum designof any structure are: structural modeling, optimum designmodeling, and the optimization algorithm. Since the structureis statically determinate, the cantilever wall does not presentany problem as far as the structural analysis is considered. Forstructural design details, the requirements of the AmericanConcrete Institute (ACI) (1990) are used. For the optimumdesign modeling, one has to study the problem parameters in­depth, so as to decide on design parameters, design variables,constraints, and the objective function. These and the design­optimization methodology are discussed in the following sec­tions.

The first four design variables are related to the geometryof the cross section, and the last three consider various steelareas. The height of the stem and the stem thickness at the topare included in the design parameters. Design parameters arepreassigned at the beginning of the structural optimization pro­cess. Other design parameters include some soil properties,unit cost of materials, loading characteristics, and others (dis­cussed later in relation to the sensitivity analysis).

X, = Total base widthX2 = Toe projectionX3 = Stem thickness at the bottomX4 = Thickness of base slabX~ = Vertical steel area of the stem per unit length of the

wallX6 = Horizontal steel area of the toe per unit length of the

wallX7 = Horizontal steel area of the heel per unit length of the

wall

Design Variables

The design variables chosen for the formulation are relatedto the cross-sectional dimensions of the wall and various re­inforcing steel areas (Fig. 1). Seven design variables are takeninto consideration. These include the following:

The main design philosophy covering the requirements,based on the behavior of reinforced concrete-cantilever retain­ing walls, is summarized in the preceding sections (Table 1).These requirements represent the failure modes that are ex-

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TABLE 4. Optimum Values of Design Variables for Example 1 TABLE 7. Input Parameters for Example 2

SOLUTION AND RESULTS

TABLE 5. Values of Behavioral Constraints at Optimum Val­ues of Design Variables for Example 1

Input parameter Unit Symbol Value(1 ) (2) (3) (4)

Height of stem m H 4.5Top thickness of stem m t 0.25Yield strength of reinforcing steel MPa h 400Compressive strength of concrete MPa f; 21Wide beam shear strength of concrete MPa {}e 0.65Concrete cover cm de 7Maximum steel percentage - Pmax 0.016Minimum steel percentage - Pmin 0.00333Shrinkage and temporary reinforcement

percent - PST 0.002Diameter of bars cm <1>..... 1.4Surcharge load kPa q 30Backfill slope degree 13 15Internal friction angle of retained soil degree <1> 36Internal friction angle of base soil degree <1>' 34Unit weight of retained soil kN/m' 'I, 17.5Unit weight of base soil kN/m' 'I: 18.5Unit weight of concrete kN/m' 'Ie 23.5Cohesion of base soil kPa c 100Design load factor - LF 1.7Depth of soil in front of wall m D 0.75Cost of steel $/kg C, 0.40Cost of concrete $/m' C e 40Factor of safety for overturning stability - No 1.5Factor of safety against sliding - N, 1.5Factor of safety for bearing capacity - SF 3.0

The program constructs an unconstrained minimization prob­lem by using a selected penalty-function approach. Four dif­ferent types of penalty functions are provided in the program:one pass-external function, the Fiacco-McCormick combinedexternal and internal function, Powell's function, andSchuldt's function (Siddal 1992).

The fonnulation and solution algorithm is tested on severalexample problems. Two representative cases are reported. Thedetails of these examples are given in the input data (see Ta­bles 3 and 7). In the example problems Sl units are used. Ineach case, both weight and cost minimization is considered.Optimum design results are shown in Tables 4-6 and Tables8-10. The optimum values of the design variables are tabu­lated together with suggested (ACI 1990; Gaylord and Gaylord1990; and Bowles 1988), upper and lower limits for easy in­terpretation (Tables 4 and 8). The values of the behavioralconstraints at the optimum and for each case are given in Ta­bles 5 and 9, respectively. Finally, the optimum values of theobjective functions, the weight of steel, and the volume ofconcrete in these designs are tabulated in Tables 6 and 10,respectively.

(4)

(5)

(00)j = 1, ... , 10

f(x) = W" + l00V/yc

Optimum Optimumvalues values

Design Lower Upper minimum minimumvariables Unit bounds bounds cost weight

(1 ) (2) (3) (4) (5) (6)

XI m 1.309 2.333 1.578 1.574X2 m 0.436 0.778 0.436 0.441X, m 0.200 0.333 0.258 0.200X4 m 0.273 0.333 0.273 0.273X, cm2/m 4.129 41.173 12.574 21.072X. cm2/m 6.551 41.173 6.551 6.551X7 cm2/m 6.551 41.173 6.551 6.681

where 'Yc = unit weight of concrete, and 100 is used for con­sistency of units.

The optimum design problem is now posed as: find x, sat­isfying

where C, = unit cost of steel; Cc = unit cost of concrete (Cc

has been selected to account for fonnwork, placing of con­crete, vibration, and so on, including the labor charges); Wst =weight of steel per unit length of the wall; and Vc = volumeof concrete per unit length of the wall.

For weight optimization the objective function definition is

Minimum Minimumcost weight

Constraint Symbol Unit value value(1 ) (2) (3) (4) (5)

Shear capacity of stem gl(X) kN 55.893 17.985Moment capacity of stem g2(X) kN'm 0 0Overturning stability g,(X) kN'm 73.333 74.001Sliding stability g.(X) kN 69.823 69.212No tension in foundation g,(X) m 0 0Bearing capacity g.(X) kPa 104.663 102.204Shear capacity of toe g,(X) kN 51.980 49.76Moment capacity of toe g.(X) kN'm 27.165 26.455Shear capacity of heel g9(X) kN 60.808 61.734Moment capacity of heel glO(X) kN'm 2.102 0

TABLE 6. Optimum Values of Objective Functions, Weight ofSteel, and Volume of Concrete for Example 1

Weight of Volume ofOptimum steel concrete

Objective function Unit value (kg/m) (m3/m)(1 ) (2) (3) (4) (5)

Minimum cost $/m 82.474 60.410 1.118Minimum weight kglm 2,498.774 79.788 1.029

[xi(Xt)max] - 1 :S 0, k = I, ... , 7 (6c)

minimizing f(x) = C,WS1 + CcVc for cost or, minimizing f(x)= Wst + 100Vc'Yc for weight.

This is a constrained nonlinear programming problem. Forthe numerical solution, a special program (RETOPT), the de­tails of which are given in Sanba~ (1995), has been prepared.

[(X;)mJnlx/] - 1 :S 0, i = 1, ... ,7 (6b)ANALYSIS

Example 1

As shown in Table 4, of the first four design variables thatdescribe the shape of the optimum wall, only the thickness of

TABLE 8. Optimum Values of Design Variables for Example 2

Optimum Optimumvalues values

Design Lower Upper minimum minimumvariables Unit bounds bounds cost weight

(1 ) (2) (3) (4) (5) (6)

X, m 1.964 3.500 2.254 2.238X2 m 0.655 1.167 0.655 0.655X, m 0.250 0.500 0.417 0.300~ m 0.409 0.500 0.409 0.409X, cm'/m 5.761 67.680 23.475 41.626X. cm'/m 11.059 67.680 11.059 11.059X7 cm2/m 11.059 67.680 11.059 11.059

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the stem at the bottom, X3, differs slightly in the two optimi­zation models (others being almost the same). As for the de­sign variables representing steel areas, while X6 resumes thelower limit for each objective function, X7 takes the lower limitfor cost optimization and is a slightly larger value than thelower limit for weight optimization. The only significant dif­ference between the results for the two objective functions isexperienced in X" the vertical steel area of the stem per unitlength of the wall. This is 12.574 and 21.072 for cost- andweight-optimization models, respectively. As for the con­straints, in Table 5, the second and fifth constraints corre­sponding to moment capacity of the stem and no tension inthe foundation and the tenth constraint corresponding to mo­ment capacity of the heel are all active in both models, ig­noring the small slack in the tenth constraint for cost optimi­zation. The optimum values for total cost and total weight are

TABLE 9. Values of Behavioral Constraints at Optimum Val­ues of Design Variables for Example 2

Minimum Minimumcost weight

Constraint Symbol Unit value value(1 ) (2) (3) (4) (5)

Shear capacity of stem g,(X) kN 75.661 0Moment capacity of stem g,(X) kN'm 0 0Overturning stability g3(X) kN'm 228.051 229.756Sliding stability g4(X) kN 142.246 143.491No tension in foundation gs(X) m 0 0Bearing capacity g.(X) kPa 1,762.754 1,756.297Shear capacity of toe g7(X) kN 42.701 36.989Moment capacity of toe g.(X) kN'm 67.200 68.555Shear capacity of heel g9(X) kN 76.315 78.083Moment capacity of heel glO(X) kN'm 13.834 1.542

TABLE 10. Optimum Values of Objective Functions, Weight ofSteel, and Volume of Concrete for Example 2

Weight of Volume ofOptimum steel concrete

Objective function Unit value (kg/m) (m'/m)(1 ) (2) (3) (4) (5)

Minimum cost $/m 189.546 155.486 2.423Minimum weight kglm 5,280.961 218.319 2.154

TABLE 11. Input Parameters for Various H-tComblnatlons

Input parameter Unit Symbol Value(1 ) (2) (3) (4)

Yield strength of reinforcing steel MPa h 400Compressive strength of concrete MPa f; 21Wide beam shear strength of concrete MPa {)e 0.65Concrete cover em de 7Maximum steel percentage - P""", 0.016Minimum steel percentage - Pmln 0.00333Shrinkage and temporary reinforcement

percent - PST 0.002Diameter of bars em <l>b" 1.4Surcharge load kPa q 25Backfill slope degree I) 10Internal friction angle of retained soil degree <I> 36Internal friction angle of base soil degree <1>/ 0Unit weight of retained soil kN/m3 "Y, 17.5Unit weight of base soil kN/m3 "Y; 18.5Unit weight of concrete kN/m3

"Ye 23.5Cohesion of base soil kPa c 125Design load factor - LF 1.7Depth of soil in front of wall m D 0.75Cost of steel" $/kg C, 0.40Cost of concrete $/m3 C e 40Factor of safety for overturning stability - No 1.5Factor of safety against sliding - N, 1.5Factor of safety for bearing capacity - SF 3.0

652/ JOURNAL OF GEOTECHNICAL ENGINEERING I AUGUST 1996

shown in Table 6. With respect to the weight of steel andvolume of concrete for each optimization model (Table 6), theformer one takes greater value for the minimum weight model,and the latter takes greater value for the minimum cost model.

Example 2

Table 8 shows the optimum values of the design variables.The variables, X2 , X.., X6 , and X7 , resume the lower limits forboth objective functions. The others fall within the lower lim­its and the arithmetic average of the lower and upper limits,as in the first example. And, X3 violates this pattern by takinga value jn between the arithmetic average and the upper limit.As in the first example, a significant difference is seen in X"which is 23.475 for the minimum cost model and 41.626 forthe minimum weight model. Design variable X3 shows a sim-

350

300

I250

!J200

__ toO.2Om

:E __toO.25m

i __toO.3Om

I 150::0...t~ 100'0

!50

03 3.5 .. ".5 5 5.5 6

Height of the Stem (Homl

FIG. 2. Effect of Height of Stem on Optimum Value of Cost·Mlnlmlzatlon-objectlve Function for Various Top Thicknesses

10000

9000

~ 8000:!!.2!

f 7000

I 6llOO.5~.a 5000

j «lOll

i 3000

J;'0 2000 __t=a.2Om'0J ___ toO.25m

~ 1000 __t=a.3Om

03.0 3.5 ".0 ".5 5.0 5.5 6.0

Height of the stem lKoml

FIG. 3. Effect of Height of Stem on Optimum Value of Welght­Minimization-Objective Function for Various Top Thicknesses

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Har tendency. It is 0.417 and 0.300 for minimum cost andminimum weight models, respectively. The second, fifth, andtenth constraints are active for each optimization model (Table9). Another constraint that is active is constraint number one,however only for the minimum weight model. Table 10 givesthe optimum value for total cost and total weight, together withthe weight of steel and the volume of concrete for each ob­jective function. The steel weight is higher in the minimum

weight model compared to the minimum cost model. The con­trary is true for the concrete volume, which is higher in theminimum cost model. This is also seen in the first example.

SENSITIVITY ANALYSIS

The design parameters considered in the present study covera wide range of parameters that are related to loading, ge-

TABLE 12. Optimum Values of Design Variables In Minimum Cost- and Minimum Weight-Optimization Models for Various H-tCom­blnatlons

H(m)(1 )

Optimum Value for Minimum Cost

t= 0.20 m I t= 0.25 m I t= 0.30 m(2) (3) (4)

(a) = XI (m)

Optimum Value for Minimum Weight

t=0.20 mit=0.25 mit=0.30 m(5) (6) (7)

3.0 1.607 1.607 1.609 1.595 1.602 1.6093.5 1.851 1.851 1.853 1.841 1.846 1.8524.0 2.095 2.096 2.097 2.087 2.090 2.0944.5 2.340 2.340 2.341 2.332 2.334 2.3365.0 2.584 2.583 2.583 2.576 2.577 2.5785.5 2.827 2.826 2.825 2.818 2.819 2.8206.0 3.070 3.068 3.067 3.061 3.061 3.062

(b X2 (m)

3.0 0.481 0.477 0.443 0.436 0.436 0.4363.5 0.522 0.517 0.514 0.509 0.509 0.5094.0 0.582 0.582 0.582 0.582 0.602 0.6034.5 0.655 0.655 0.655 0.658 0.684 0.7005.0 0.736 0.730 0.727 0.735 0.762 0.7895.5 0.826 0.819 0.814 0.812 0.839 0.8666.0 0.921 0.914 0.908 0.889 0.916 0.943

(c) X, (m)

3.0 0.272 0.272 0.300 0.200 0.250 0.3003.5 0.313 0.312 0.312 0.212 0.250 0.3004.0 0.355 0.354 0.354 0.244 0.250 0.3004.5 0.399 0.399 0.399 0.278 0.278 0.3005.0 0.447 0.446 0.445 0.316 0.316 0.3165.5 0.495 0.495 0.494 0.356 0.356 0.3566.0 0.546 0.545 0.545 0.400 0.400 0.400

(d) X. (m)

3.0 0.273 0.273 0.273 0.273 0.273 0.2733.5 0.318 0.318 0.318 0.318 0.318 0.3184.0 0.364 0.364 0.364 0.364 0.364 0.3644.5 0.409 0.409 0.409 0.409 0.409 0.4095.0 0.455 0.455 0.455 0.455 0.455 0.4555.5 0.500 0.500 0.500 0.500 0.500 0.5006.0 0.545 0.545 0.545 0.545 - 0.545 0.545

3.0 13.293 13.212 11.404 25.341 15.392 11.4043.5 16.011 16.032 16.051 37.382 23.857 17.1084.0 18.905 18.942 18.916 39.673 36.778 24.8354.5 21.852 21.889 21.861 42.319 42.319 35.5025.0 24.829 24.856 24.924 45.123 45.123 48.0095.5 27.977 28.007 28.035 48.009 48.009 48.0096.0 31.224 31.257 31.288 50.942 50.942 50.942

3.0 6.518 6.518 6.518 6.518 6.518 6.5183.5 8.031 8.031 8.031 8.031 8.031 8.0314.0 9.545 9.545 9.545 9.545 9.545 9.5454.5 11.059 11.059 11.059 11.059 11.059 11.0595.0 12.572 12.572 12.572 12.572 12.572 12.5725.5 14.086 14.086 14.086 14.086 14.086 14.0866.0 15.600 15.600 15.600 15.600 15.600 15.600

3.0 6.518 6.518 6.518 7.499 7.039 6.5773.5 8.031 8.031 8.031 9.078 8.689 8.1884.0 9.545 9.545 9.545 10.669 10.374 9.8094.5 11.059 11.059 11.059 12.358 12.041 11.5885.0 12.572 12.572 12.572 14.136 13.784 13.4355.5 14.086 14.086 14.086 16.016 15.630 15.2476.0 15.600 15.600 15.600 17.991 17.570 17.152

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5400

5000

5200

5800

5600

4800

220 -,------------.,.- 6000

FIG. 6. Effect of Internal Friction Angle of Retained 5011 on Op­timum Values of Objective Functions for Minimum Cost- andMinimum Weight-Optimization Models

100 -j----t-----t---+---+ 4600

28 30 32 34 36Internal Friction Angle or Retained Soil (degree).- -..

-+-Min.Cost (gm)

__Min'wcight (kg/m)

iables for the rest of the design parameters are briefly illus­trated.

The input parameters of the wall considered for the sensi­tivity analysis for height and top thickness are given in Table11. Both objective functions are considered, and their optimumvalue variations with respect to changes in height of the stemfor various top thickness values are given in Figs. 2 and 3.Table 12 shows the optimum values of the design variablesfor both the minimum cost- and minimum weight-optimizationmodels. In the range of the stem height considered (3-6 m),Figs. 2 and 3 reveal that for higher values of height, the op­timum cost and weight become more sensitive to variations inthe stem height. This is more apparent for the cost function.For example, when t = 0.20 m and H changes from 3.0 to 6.0m, the cost of the wall increases 3.73 times, whereas theweight increases 3.38 times. These rates show a decliningtrend as t increases from 0.20 to 0.30 m. Smaller top thickness

. values of the stem produce more favorable optimum solutionsfor both objective functions.

With regards to the design variables (Table 12), the opti­mum values of the first four are sensitive to changes in H forboth of the minimization models. For a given H, the first fourdesign variables are not much affected by increases of t, fromt = 0.20 to 0.30 m, apart from the third design variable (stemthickness at the bottom), the lower bound of which is t. Forboth minimization models, the optimum values of the lastthree design variables corresponding to reinforcing steel areasshow sensitivity to changes in H, but not in general to shiftsin t. Only the vertical steel reinforcing area in the weight­minimization model is influenced by changes in t.

The input parameters assumed for the study of sensitivityfor surcharge loads are mostly the same as for the sensitivityanalysis for Hand t (Table 11). Though here, H = 4.5 m, t =0.25 m, "Is = 18 kN/m3

, and D = 1.0 m. The optimum valuesfor the objective functions are shown in Fig. 4 for the sur­charge load, varying from 0 to 50 kPa. According to Fig. 4,the cost-minimization model is more sensitive to variations insurcharge load compared to the weight-minimization model.In fact, as q changes from 0 to 50 kPa, the optimum costincreases 1.32 times, and the optimum weight increases by1.20.

As for the sensitivity of the design variables, significant sen­sitivity is observed in X I -X3 (total base width, toe projection,and stem thickness at the bottom) and Xs (vertical steel areaof the stem). These variables increase as the surcharge load

·§ 215 5800.51:l!! 205 5600...,:;

~195 5400

a185 5200.. ""

J)~ a 175 5000,seII ... 165 4800~f.';1!~ 155 4600

i 145 4400~0'C 135 4200....;;> 125 4000

0 10 20 30 40 50

Surcharge Load (q-kPa) -+-Min.Cost ($1m)

__Min'wcight (kg/m)

.il 200 5500

.!l)l 5400....iii 190i 5300~

1.-

~~ 180 5200

.. aa~ 5100

if. 170!~ 5000..~ 4900"E 160Q

os 4800..oj> 150 4700

0 5 10 15 20 25 30__Min. Cost (gm)

BlII:kflll Slope (degree) __ Min. Weight (kg/m)

FIG. 4. Effect of Surcharge Load on Optimum Values of ObJec­tive Functions for Minimum Cost- and Minimum Weight-Optimi­zation Models

FIG. 5. Effect of Backfill Slope on Optimum Values of ObJec­tive Functions for Mlnlmum-Cost- and Minimum Welght-optlml­zatlon Models

ometry, soil properties, code specifications, unit cost, and othercharacteristics of construction materials. Sensitivity of the op­timum solution to changes in these parameters is an importantissue as far as practical design is concerned. The analysis re­sults include the sensitivities of the optimum weight and op­timum cost as objective functions and the optimum values ofthe seven design variables. As a representative of such anal­yses, results concerned with the sensitivity of optimum solu­tions with respect to height and top thickness of stem, sur­charge load, backfill slope, internal friction angle of retainedsoil, and the yield strength of reinforcing steel are reported.Sensitivities of the objective functions are explained for alldesign parameters considered. However, due to space restric­tions, from the sensitivities of the design variables, only thoserelated to changes in height and the top thickness of the stemare reproduced (Table 12). The sensitivities of the design var-

6541 JOURNAL OF GEOTECHNICAL ENGINEERING 1AUGUST 1996

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175 -1-------+--------;275 345 400f,-Yield Strength of Reinforcing Steel (MPa)

FIG. 7. Effect of fyand f'; on Optimum Value of Cost-Mlnlmlza­tion-Objectlve Function

variables show some degree of sensitivity, though not verysignificant.

For the sensitivity analysis related to the internal frictionangle of retained soil, the same parameters are taken as in thecase of the surcharge load analysis. However, two additionalparameters, q = 15 kPa and J3 = 10° are added. The optimumsolutions for weight and cost are shown in Fig. 6. Both modelsare sensitive to changes in the internal friction angle of re­tained soil. The minimum weight model is more sensitive inthe range considered when compared to the minimum costmodel.

As far as the design variables are concerned, in the cost­minimization model, X4 (thickness of base slab), X6 (horizontalsteel area of the toe), and X7 (horizontal steel area of the heel)are insensitive to changes in the internal friction angle, andothers decrease as the angle increases in the defined range. Inthe weight-minimization model, X4 and X6 are insensitive tochanges in the internal friction angle, and the others decreasewith an increase in the angle, with the exception of X5 (verticalsteel arc of the stem). And, X5 shows a decrease in cost min­imization and an increase in weight minimization with an in­crease in the angle.

The input parameters for studying sensitivity with respectto yield strength of reinforcing steel are the same as those forthe internal friction angle, with <l> = 36° added to the list. Theoptimum weights and costs are shown in Figs. 7 and 8. Boththe cost and weight are reduced as the reinforcing steelstrength is increased. This is more significant for the cost­optimization model. For example, with respect to the com­pressive strength of concrete of 21 MPa, using fy = 400 MPainstead ofh =275 MPa reduces the cost by 10%.

For both models, the design variables that are notably sen­sitive to increases in the yield strength are those correspondingto the reinforcing areas-the most sensitive being the verticalreinforcement area.

ACKNOWLEDGMENTS

CONCLUSIONS

This paper discusses optimum design of reinforced con­crete-cantilever retaining walls as a part of a comprehensiveinvestigation of retaining structures. The problem is posed asa nonlinear programming problem. In this formulation, twoobjective functions, the cost and weight of the wall, are con­sidered. The constraints are the design requirements that arethe behavioral constraints and side constraints defined as lowerand upper limits of the design variables. The optimum solutionis obtained using a specially prepared computer program(RETOPT). 1\\10 numerical examples are given, and the resultsare discussed. The proposed method automatically yields anoptimum design in terms of seven design variables coveringcertain dimensions of the cross section and reinforcing areas.The formulation is quite flexible and open to further modifi­cations and extensions. Additionally, sensitivity of the opti­mum solutions with respect to certain design parameters canbe performed, also illustrated with numerical examples.

APPEND~. REFERENCES

This paper is part of an MS thesis entitled, "Optimum Design of Re­inforced Concrete Cantilever Retaining Walls by Non-linear Program­ming," submitted to the Department of Civil Engineering, Middle EastTechnical University, Ankara, Turkey. The thesis was prepared by thefirst author under the supervision of the second author. The contributionsof J. E. Bowles and Prof. A. V. Keskar during the preparation of thethesis are gratefUlly acknowledged.

Alshawi, F. A. N., Mohammed, A. I., and Farid, B. J. (1988). "Optimumdesign of tied-back retaining walls." The Struct. Engr., London, En­gland, 66(6), 97-105.

JOURNAL OF GEOTECHNICAL ENGINEERING / AUGUST 1996/655

__ fc=21 Mr.

___ fc=24 Mr.

__fc=28Mr.

5120

5100

200

E -+- rc=21 MPa~

.E __rc=24MPac 195:i .....-rc-28MPa...0-c~

190c~-II. E~~'filiiIII 0:soU

1850....0III~

~E 180~

.Ea.0

5140 -,---------------,

'0!~ 5040E~

.Sg 5020 -f--------+---------i

275 345 400f,-Yield Strength of Reinforcing Steel IMPe'

FIG. 8. Effect of fyand f'; on Optimum Value of Welght-Mlnlml­zation-Objectlve Function

increases, in both of the optimization models. The optimumvalue of X7 (horizontal steel area of the heel) also increaseswith an increase in the surchage load, yet, after 20 kPa andonly in the weight-minimization model.

The input parameters considered for sensitivity, with respectto backfill slope, are similar to those used for the study of theeffect of the surcharge load with an added parameter q = 15kPa. The range of backfill slope conceived is from 0° to 30°.The optimum weights and costs are shown in Fig. 5. Objectivefunction values gradually decrease as the backfill slope in­creases from 0° to 20°, and then they start increasing again,approaching towards a slope of 30°.

Reflecting on the optimum design variables, X4 (thicknessof base slab), X6 (horizontal steel area of the toe), and X7

(horizontal steel area of the heel) are insensitive to changes inbackfill slopes in both minimization models. The other design

:c'"~E=E'c:i....2c

.S!ti_~.g, 5080~:!.

tl~ 5060o

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"Building code requirement for reinforced concrete, with design appli­cations." (1990). ACI 318 -89, American Concrete Institute (ACI), De­troit, Mich.

Bowles, J. E. (1988). Foundation analysis and design, 4th Ed., McGraw­Hill Book Co., Inc., Singapore.

Dembicki, E., and Chi, T. (1989). "System analysis in calculation ofcantilever retaining walls." Int. J. Numer. and Analytical Methods inGeomech., 13, 599-610.

Fang, H. Y., Atsuta, T., and Chen, W. F. (1980). "Optimum design ofretaining walls on sloping hillside." 6th Southeast Asian Con! on SoilEngrg., Taipei, 391-406.

Gaylord, E. H. Jr., and Gaylord, C. N. (1990). Structural engineeringhandbook, 3rd Ed., McGraw-Hill Book Co., Inc., New York, N.Y.

656/ JOURNAL OF GEOTECHNICAL ENGINEERING / AUGUST 1996

Keskar, A. V., and Adidam, S. R. (1989). "Minimum cost design of acantilever retaining wall." Indian Concrete J., Bombay, India, 401­405.

Pochtrnan, Y. M., Zhmuro, O. V., and Landa, M. S. (1989). "Design ofan optimal retaining wall with anchorage." Soil Mech. and Found.Engrg., 25(5), 508-510.

Rhomberg, E. J., and Street, W. M. (1981). "Optimal design of retainingwalls." J. Struct. Div., ASCE, 107(5).

Sanba§, A. (1995). "Optimum design of reinforced concrete cantileverretaining walls by non-linear programming, MS thesis, Dept. of Civ.Engrg., Middle East Tech. Univ., Ankara, Turkey.

Siddall, J. N. (1982). Optimal engineering design. principles and appli­cations. Marcel Dekker, Inc., New York, N.Y.

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