j.c.perezzan & s. hernandez - wit press...size and shape optimization of long portal bridges...

Size and shape optimization of long portal

bridges

J.C.Perezzan & S. HernandezE. T.S Ingenieros de Caminos, Canales y PuertosUniversity of La CorunaCampus de Elvina, La Coruna 15071, SpainEmail: [email protected];[email protected]

Abstract

Most applications in optimum design of frame structures have been devoted tosize optimization. In this paper a multispan bridge having a large number ofelements is presented as a case for design optimization. First, a size optimizationproblem is carried out having a number of design variables varying from 4 to118. Then size and shape optimization is formulated considering the span lengthas design variable for geometry optimization. Numerical results obtained showthe advantages of this combined approach.

1 Optimum design of flexural systems

Optimum design of flexural systems composed by bar elements is a very wellstablished topic on structural optimization. Plastic and elastic behaviour has beenconsidered for many authors using classical mathematical programming oremerging techniques as genetic algorithms or neural networks. References [1-11]represent an overview of results in different problems and typologies includingframes, grillages and beams.

While the number of applications is large, all of them are related with sizeoptimization. The design problem is posed usually in terms of the cross sectionareas of the bars which are considered design variables. Other mechanicalparameters, as inertia modulus /, or strength modulus W are linked to area A byusing the following expressions

7 = a,A*' W=a,,A^ (1)

Transactions on the Built Environment vol 37 © 1999 WIT Press, www.witpress.com, ISSN 1743-3509

288 Computer Aided Optimum Design of Structures

Sometimes, design optimization is made in several steps by carrying outmultilevel optimization techniques [12-13]. In that approach, the optimumvalues of the areas are first identified, then a second optimization problem issolved considering the internal dimensions of the cross section areas of the bar asdesign variables.

In this paper an application of size and shape optimization of portal bridgeswill be presented. The set of design variables will be composed by the values ofcross section areas A defined as a vector /4_, and the set of bars length L asindicated in Figure 1. Allowable stresses and vertical displacements will beadopted as constraints and the weight of material will be considered as objectivefunction. Hence, minimization problem may be written as

^

i, h

Figure 1. Size and shape of a portal bridge

K, = W, (X) (2.a)

being X the vector of design variables

X = %f&W (2.b)subjected to

a^<a(X)<(Tu (2.c)wW<%(/ (2.d)

whereWg weight of materialA. vector of cross section areasL vector of bars length<7 representative stress in a structural element0%, <j(/ lower and upper stress level of materialu representative vertical displacement of a structural elementUy upper bound of vertical displacement

2 Sensitivity analysis approaches

Efficient numerical optimization algorithms require first order information of theobjective function and the set of constraints of the design problem. Such kind ofinformation is known as sensitivities. First order sensitivity of a structuralresponse \// with respect to a design variable *, can be written as

dy _ dy/ ^¥_d^ (3)dx dx du dx


Computer Aided Optimum Design of Structures 289

It is well know that expression (3) may be solved by two approaches.In the direct differentiation method the following expression is finally

obtained

f- ——Aou

a)

b)

Alternatively, the adjoint approach leads to

doc(5)

Both techniques require to obtain the following:

Derivatives of vector of loading .~£.: The main class of loading depending

on the design variables is the self weight of the structure. This term isusually neglected in truss systems but it may be important in most flexuralsystems and it will be included in this paper.

Derivatives of stiffness matrix ^A: Derivatives of the stiffness matrix withdx,

respect to design variables as cross section area or other parameter related toit, as inertia modulus, is quite straight forward. But major difficulties appearif coordinates of nodes are included in the set of design variables. In thiscase the derivatives usually are not evaluated explicitly, and instead that arecarried out by finite differences procedures. In the piece of research presentedin this paper the derivatives of the stiffness matrix was done in closed form.Optimization was carried out using ADS [14] software using sensitivitiesprovided by the authors for both types of design variables: cross sectionareas and nodes coordinates.

3 Structural optimization of long portal bridges

The example included in this paper is a long bridge composed by twenty fourspans of 93.75 m of length and forty and three spans 250 m long each one.Because the symmetry only half of the bridge is shown in Figure 2.

L, 1 i ,

i i

, l4020

95.75 9J.75 250 725#72 #22

7725 ' 5J75

Figure 2. Geometry and loads considered



The design is similar to the Confederation bridge recently built in Canada[15-16]. The actual bridge was designed in concrete but in this paper steel andcomposite cross sections were considered as indicated in Figure 3.

&2J 11

a) Columnsbl) Actual shape b2) Transformed shape

b) DeckFigure 3. Cross sections considered

Defining / as inertia modulus and Wj, W/2, as the strength moduli corres-ponding to depth hj and /%, respectively the following relationship were set up

W, = a», Abwl W2 = \bw2 (6)

Numerical values of each parameter in (6) depended on the cross sectionareas as follows.—Columns:For A> 11000cm'

aj = 2.27 • 10~*° bj = 6.94 tf^ = a^ =1.42- 1C

For A < 11000cm*5-j S-- I y-»—42 1 J \ -J<~1 f\ A -J.36-10 bf = 12.37 a^ = a^ = 9.43-1

—Bridge deck:For A > 17000 cnf

6^=6^=3.72 (7.a)

(7.b)

a =3.74-70 6 =2.&7

For A < 17000 cnfa, = 2.99-JO-'* bj = 5.21

a, =7.73-70 6_ =2.40

= 4 J2 • 7

, = 2. 2

= 4.23 • 1

(7.c)

(7.d)

The set of loads considered were self weight of the structure and a distributedload p = 9.36 t/m, equivalent to a service load of 0.4 t/nf on bridge deck.

3.1 Size optimization

In size optimization only cross section areas were considered as design variables.Three cases were examined.1) Four cross section areas: In this design the shorter spans and columns had

constant area A/, A* respectively and the longer spans had two different areasize for the deck, namely A^ A,, as indicated in Figure 4a.



^

n<

^"-1<

**-l<

"~-y<

1

^ K 3K

" ^

Q ^D _^?u "

1 i§ t

-# Jh

< 1% 3 <

k fN

> JT< ^ ^k

1?K

P:k

<" p.1% ^

^ ^

N %^< <^

k

PK

$k

t\ QO< < p

Ys

VV

ftfc

pk

t\ OQ< <K

k

6) <$ design variable case

T"

-C5

i

sri

V-i-qT

iQ\ofN<txr00«N<<IX«N5<•O<N<<"<rr,T

0

J

J

K oo*?*?&

k

tx oo^<TK

fc

N OD

&

10 OT T F

K

rn Xh<r <r K

^r^pK

^^rpk

^ PK

^s

3•2

00•*~

Figure 4. Design variables in size optimization



2) Eight cross section areas: In this case the longer spans had four crosssection sizes, A?, A# A$, A& the shorter spans had two different areas Ay,A2, and also columns had two areas A 7, A&

3) A hundred and eighteen areas: The distribution of area variables is indicatedin Figure 4c.For each design case two different problems were considered depending on

the class of constraints included:a) Stress constraints: Allowable stress level, Oy = -o^ = 2600 Kp/cnf.b) Displacement constraint: Maximum vertical displacement //250, being /

any span length.The optimum values of the objective function are presented in Figure 5.

2-

1 --

-Q- stress & displ. constraints-o- stress constraints

1.63

1.260.94

-t-118Number

of variablesFigure 5. Evolution of objective function values

Optimum values of the objective function and the set of design values foreach case appear in Tables 1 to 3.

Table 1. Four size variables case

Constraints

Stress

Stress & Displ

Ay fern')

9182

9231

A, fern';

27234

35620

A, (cm')

18660

24022

AXcm'j

14818

18536

W (fgMO*

1.28

1.63

Table 2. Eight size variables case

Constraints

Stress

Stress & Displ

Constraints

Stress

Stress & Displ

A, (cm")

10580

9142

A, (cm')

11414

18799

A, (cm')

7950

9296

A 7 (cnf)

13714

18800

A, fern")

25390

36577

A, fern")

9516

12401

A, fern")

21314

31937

W (Kg)* 10"

1.07

1.54

A, (cm')

17114

26575



Table 3. A hundred and eighteen size variables case

ConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & Displ

A, (cm")90658488

A, (cm")79507982

A,, (cm")79507970

Ajy (cm*)79877985

A# fern")2191233025

A 3, (cm*)1472723534

A,; (cm")2257932806

A,, (cm")1430322914

A,, (cm")2274633010

A,, (cm")1378920656

A,, (cm";2539041846

A,; (Cm")1472522712

A 7, (cm*)2256132765

A,, (cm")1431522784

A,, (cm")2258632773

A, (cm")83509249

A, (cm")79507950

A,, (cm")79507950

A,, (cm')79507950

A,, (cm")1777628198

A,, (cm?)824715692

A,* (cm*)1849928194

A,, (cm")853915088

A so (cm*)1880028652

A,, (cm")931416193

A,, (cm")2131435929

AM (cm")1087615226

A,, (cm")1846528116

A** (cm")858615000

AM (cm")1851028134

L A, (cm")84608488

Ay (cm")79507970

A,, (cm")• 7950

7970A,, (cm")79508045

A,; (cm")1568623088

A,, (cm")2253132730

A,, (cm")1429122855

A,, (cm")2370832639

A,, (cm")1465423508

A,; (cm")2504239044

A,, (cm")1711429236

A«y (cm")2271432801

A;, (cm")1426122747

A,, (cm")2258432770

A,; (cm")1430922773

A, (cm")79508048

A™ fern")79507950

A,, (cm")79507950

A,, (cm")79507950

A,, (cm")1203818719

A,, (cm")1841828016

A,, (cm")859715025

A,, (cm")1975027906

A,, (cm")841815171

A,, (cm")2216535940

A,, (cm")1141519391

A?, (cm")1874828184

A;, (cm")864315008

A,, (cm")1850828127

AM (cm")860014995

A, (cm")79507982

A,, (cm")79507970

A/7 (cm")79507985

A,, (cm")1239712246

A,, (cm")2282733096

A,, (cm")1421922604

A,, (cm")2258432846

A,7 (cm")1427022529

A,, (cm")2222831887

AM (cm")1861531644

A,5 (cm")2221832810

A,, (cm")1457722843

A 77 (cm")2258932777

A,, (cm")1430622763

A,, (cm")2258632771

A, (cm")79507950

A,, (cm")79507950

AM (cm")79507950

A,, (cm")86007950

A jo (cm")1891928689

A,, (cm")868515138

A,, (cm")1851528245

A,, (cm")977215291

A,, (cm")1788626418

A,, (cm")942323665

A,, (cm")1794328122

A* (cm")837015073

A 7* (cm")1851828142

A*, (cm")860014994

A,,, (cm")1851028131



Table 3. A hundred and eighteen size variables case (cont.)

ConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & Displ

Ayj (cm*)1431022768

Ayj (cm*)2258632772

A 103 ( )1431122770

A 109 (cm*)2398834674

AJJS (cm*)1238115911

Ay; (cm*)860114994

AM (cm*)1851128132

AJM (cm*)860414994

A jia (cm*)1991230034

A ii6 (cm*)951611179

AM (cm*)2258632772

Ayy (cm*)1431122769705 (cm*)2258632772

• in (cm*)1571224672

A,,7 (cm*)1251711451

A^ (cm1851128133

AJM (cm*)860314994

Aio6 (cm*)1851128132

A, ,2 (cm*)1000916897

Aj,s (cm*)95167478

A^ (cm*)1431022770

A,OJ (cm*)2258632772

AJM (cm*)1431122769

Aj,3 (cm*)95166515W*10*0.941.26

Ayt (cm*)860314994

4 102 (cm*)1851128132

A 108 (cm*)860414994

A, „ (cm*)95166574

3.2 Size and shape optimization

When shape optimization was considered in the problem the length of spans wereincluded as design variables. In this way variables L, to L^ represented length ofshorter spans and L,? to L^ corresponded to longer spans. The total length ofeach class of spans was considered a parameter. Therefore the followingrelationship arised.

12L; = 7725

34

%4 = (8)

i=13

II 1 r > . 1 ^

13 -34#12 #22

1125 ' 5375

Figure 6. Definition of shape design variables

40

20

Design variables representing cross section areas were those aforementioned.Hence, the following cases were worked out.



Table 4. Number of design variables

Span length

343434

Cross sec

1

tion areas

4818

Total

3842152

Again, an optimization problem considering only stress constraints andanother problem including also vertical displacement constraints were carried out.In both cases the constraint bounds were those mentioned in the sizeoptimization problem.

Evolution of the objective function with regards to the number of designvariables is presented in Figure 7.

.70*1

7 0.93t-

0.910.84

-o- stress & displ constraints-o- stress constraints

0.83

0.81 0.80

750Number

of variablesFigure 7. Evolution of objective function values

Values of objective function, cross section areas and span length appear foreach case in Tables 5 to 7.

Table 5a. Size and shape optimization (38 variables case)

Constraints

StressStress & DisplConstraintsStressStress & Displ

LI (m)

80.7780.76L,(m)98.7698.57

Lt(m)98.7698.73Lg (m)98.7798.52

Lj(m)98.7798.67L,(m)98.4798.54

L<(m)98.7698.66L,o (m)96.6498.23

Lg(m)98.7798.63Lj, (m)95.1096.31

Lg(m)98.7798.59LU (m)62.6660.70



Table 5a. Size and shape optimization (38 variables case) (cont.)

ConstraintsStressStress & Displ

ConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & Displ

L,3 (m)174.45199.26

Ljy (m)273.25265.27

^25 M270.49262.11

LJI (m)250.52262.62

LU (m)255.49254.52

620 (m)273.66261.04

J-26 (m)271.90262.54

LU M249.74262.62

LJS (m)274.77261.81

LZJ (m)250.69241.89L (m)272.65262.59L (m)250.13262.62

LU (m)270.67262.44

£22 W154.38154.03

&24 )266.66262.60

6,4 (m)125.00131.30

LU (m)273.94

261.71

1-23 (m)205.75212.02

£j, (m)251.15262.61

Ljg (m)254.21248.84

1 24 W256.55257.94LM (m)249.00262.61

Table 5b. Size and shape optimization (38 size variables case)

Constraints

Stress

Stress & Displ

A, (W)

8279

8263

A, fern')

21717

21944

A, fcm')

10784

10766

AXcm'j

9516

10701

W (Kg)* 10*

0.91

0.93

Table 6a. Size and shape optimization (42 size variables case)

ConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & Displ

L,(m)79.9679.99Ly(m)98.5498.62

Lj3 (m)194.91199.30LM (m)265.54267.06

£25 M259.00263.66

LSI (™)266.19264.61

L,(m)98.7398.78

4, M98.4998.58Lj4 (m)253.08254.73

LZO M266.87261.75L (m)264.14264.45

Lj2 (m)266.38264.60

Lttm)98.6998.75

L,(m)98.5298.61LJS (m)265.91263.37LU (m)240.87237.39

4,7 M265.15264.55LJS (m)266.47264.61

L<(m)98.6598.72

LH, (m)98.2498.36Ljt (m)263.85264.32

^22 M149.56135.54

Z/2* (rn)265.52264.58Lj4 (m)133.33132.31

L,(m)98.3498.68LU (m)96.1098.58L,7 (m)264.39263.67

L23 (m)207.44207.19

Lf9 (m)265.77264.59

L,(m)98.5898.65

Lj2 (rn)61.8961.28Ljg (m)243.19250.83

L24 (m)244.73257.29

Lj,, (m)266.01264.60



Table 6b. Size and shape optimization (42 size variables case)

ConstraintsStressStress & DisplConstraintsStressStress & Displ

A, (cm")83488355

A, (cm")915811375

A, (cm*)79507950

A 7 (cm*)1070010700

A, (cm")2086022064A, ("%")1070010700

A, (cm")1408713910

W(Kg)*10*0.810.84

A, (cm")103479997

Table 7a. Size and shape optimization (152 variables case)

ConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & Displ

Lj(m)80.2080.94Lj(m)98.5696.37LJI (m)194.19208.99Ljy (m)263.33261.80£25 M259.37258.09LSI (m)266.20264.92

L,(m)98.5698.65L,(m)98.4193.71LU M256.14250.06&2o (m)261.15258.441 26 W262.96261.90Lj2 M266.32264.40

Lj(m)98.6899.59L,(m)98.9393.45LJS (m)264.10259.36l-2i (m)243.24246.791 27 M264.86263.68Lj3 (m)266.40264.04

L<(m)98.5997.80Lm (m)96.4293.14Ljg (m)264.44260.85J-22 (m)152.35165.69LZS (m)265.22264.59L (m)133.29131.96

L;(m)98.6098.41LJ, (m)95.8292.44Lj7 (m)263.14259.791 23 (m)206.60204.34&2P (m)265.66265.04

L<(m)98.5796.19Lj2 (m)63.6484.32L,g (m)243.81246.75L24 (m)246.28248.35LJO (m)265.95265.17

Table 7b. Size and shape optimization (152 variables case)

ConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & Displ

Ay (cm")83668431

A 7 (cm*)83438275

A,, (cm")83478138

A,, (cm")82057950

A,, (cm*)2045119411

A, (cm")79507951

A, (cm*)79507950

A,, (cm")79507950

A,, (cm")79507950

A,, (cm")1367812959

A, fcm^83328344

A, (cm83468300

A,, fern";83497950

A,, (cm?)82787950

A,; (cm")1039410371

A, (cm")79507950

A/o (cm*)79507950

A,, (cm")79507950

A,, (cm")79507950

A,, (cm")899710984

A, (cm")83478394

A/; (cm*)83478146

A,; (cm")83457950

A,, (cm*)828710406

A,, (cm*)2061922280

A, (cm")79507950

A,, (cm")79507950

Ajy (cm*)79507950

A*, (cm")79507950

A,, (cm")1402613459



Table 7b. Size and shape optimization (152 variables case) (cont.)

ConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & Displ8627StressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & DisplConstraintsStressStress & Displ

A,/ (cm97958716

A,, (cm")2074622703

A,, (cm*)98938509

A 4, (cm*)2074122111

ASS (cm*)98458530

A,, (WV2054721151

A,7 (cm*)1030810551

Anfwfj2067422226

A%, (cm*)96468651

A,, (cm2080422672

Ayj (cm*)98508627

A^ (cm*)2088122827

A, M (cm*)99548562

Aj (cm*)2086522750

A us (cm*)1070010700

A,, (WJ903211125

A,, (cm*)1398813527

A« (cm*)897711204

A ;„ (cm*)1404013659

AS, (cm*)887311166

A,, (cm*)1378014208

A,, (cm*)895710959

Aj4 (cm*)1404813614

Ago (cm*)878011348

A,, (cm1404013721

AM (cm*)899611355

AM (cm*)1404413682

A, 04 (cm*)904911294

Ajjo (cm*)1404013633

A,,, (cm*)1070010700

A,, (Wj2070222671

A,, (cm98348484

A,, (cm*)2087522039

ASJ (cm*)9650

• 8658A,? (cm*)2039222847

A,, (cm')1041811409

A,, (cm*)2033421785

A 75 (cm*)97468662

AW (cm*)2078522586

A*7 (cm*)97938637

A,, (cm*)2086122792

AM (cm*)99278584

A jos (cm*)2087922776

AJU (cm*)101128553

AH, (cm*)1070010700

A,, (cm2j1395313550

A,,, (cm*)891111219

A,, (cm*)1407213558

A,, (cm888611253

AS, (cm*)1404213505

A,, (cm*)87948935

A 70 (cm*)1371913454

A;, (cm877211302

A,, (cm1403613712

AM (cm*)896111362

A,, (cm1404513709

Ajoo (cm*)902511320

A 106 (cm*)1403613639

A//2 (cm*)907611279

Ajjg (cm*)1070010700

A,, (cm98288562

A,, (cm*)2080622521

A,7 (cm*)98778634

A,, (cm*)2060922438

Asy (cm*)100498875

A,, (c m')2034519504

A,, (cm*)98168723

A 77 (cm*)2070522451

A,, (cm97138645

Agy (cm*)2084022736

A,, (cm*)98968611

AJOJ (cm*)2089222773

A/07 (cm*)99398554

AH; (cm*)1070010700W*70*0.800.83

A,, (cm*)901911230

A*, (cm*)1404213520

A,, (cm*)901911410

As4 (cm*)1392213499

A*, (cm*)890011137

A 66 (cm*)1371813116

A 72 (cm*)868711165

A;, (cm1402013683

A,, (cm900511363

Ayo (cm*)1404513720

Aye (cm*)900511342

A/w (cm*)1403413654

AIM (cm*)904811282

A 1 14 (cm*)1070010700



4 Conclusions

Some conclusions may be obtained from the results presented in the paper.— Size optimization of long civil engineering structures produces a quite

important reduction on structural weight without complicating theconstruction process.

— Optimum design of frame structures has been made frequently, butapplications taking in account changes on geometry are scarce.

— Shape optimization of bridges produce even further advantages that thoseobtained by size optimization. Introducing spans length as design variablesallows the material to be distributed along the bridge in a more efficientway.

— For the example posed, optimal design with size and shape design variablesleads to almost the same solution, in terms of the structural weight, forstress constraints that for stress and displacement constraints.

5 References

1. COHN, M.Z., GHOSH, S.K. and PARIMI, S.R., "Unified Approach tothe Theory of Plastic Structures", /. Eng. Mech. Div., ASCE, vol. 98, No.EM5, October 1972, pp. 1133-1158.

2. HORNE, M.R. and MORRIS, L.I., Optimum Design of Multistory RigidFrames, in Optimum Structural Design, Gallagher, R.H. (ed), John Wiley,1973.

3. KAVLIE, D. and MOE, J., "Automated Design of Frame Structures", J.Struc. Div., ASCE, vol. 97, No. ST1, January 1971, pp. 33-62.

4. MOSES, F. and ONODA, S., "Minimum Weight Design of Structureswith Application to Elastic Grillages", Int. J. Num. Meth. Eng., vol. 1,pp. 311-331, 1969.

5. KAVLIE, D. and MOE, J., "Application of Nonlinear Programming toOptimum Grillage Design with Nonconvex Sets of Variables", Int. J.Num. Meth. Eng., vol. 1, pp. 351-378, 1969.

6. HERNANDEZ, S., Size and Shape Optimization of Truss and FrameStructures with Multiple Local Optima. 33rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference. Dallas(USA), 1992.

7. GRIERSON, D.E., Computer-Automated Optimal Design of StructuralSteel Frameworks, in "Optimization and Artificial Intelligence in Civil andStructural Engineering", B.H.V. Topping (ed.), pp. 327-354, KluwerAcademic Publishers, 1992.



8. GRIERSON, D.E. and XU, L., Design Optimization of Steel FrameworksAccounting for Semi-Rigid Connections, in "Optimization of LargeStructural Systems", G.I.N. Rozvany (ed.), pp. 873-882, Kluwer AcademicPublishers, 1992.

9. ADELI, H. and CHENG, N.T., "Concurrent Genetic Algorithms forOptimization of Large Structures", Journal of Aerospace Engineering,ASCE, Vol. 7, No. 3, pp. 276-296, 1994.

10. ADELI, H. and PARK, H.S., "A Neural Dynamics Model for StructuralOptimization - Theory". Computers and Structures, Vol. 57, No. 3, pp.383-390, 1995.

11. ADELI, H. and PARK, H.S., "Optimization of Space Structures by NeuralDynamics", Neural Networks, Vol. 8, No. 5, pp. 769-781, 1995.

12. SOBIESCZANSKI-SOBIESKI, J., Multilevel Structural Optimization,Computer Aided Optimal Design, NATO/ASI Seminar, Vol. 3, pp. 7-28,1986.

13. KIM, D., Multilevel-Multiobjective Optimization for EngineeringSynthesis, Ph. D. Thesis, University of California in Santa Barbara, 1989.

14. ADS User's Manual, VR&D, 1988.

15. GILMOUR, R., SAUVAGEOT, G. TASSIN, D. and LOCKWOOD, J.D.,Northumberland's Ice Breaker. Civil Engineering, pp. 34-38, ASCE,January 1997.

16. DUPRE, J., Bridges. Black Dog & Leventhal, 1997.


j.c.perezzan & s. hernandez - wit press...size and shape optimization of long portal bridges...

Documents