Optimization of Stiffened Plates for Steel Bridges using GeneticAlgorithms.Ricardo ParraABSTRACT: Bridges are an important part of the infrastructure of states. Welded stiffened plates are widelyused as a box girder for bridges. A design optimization problem for a single steel box with stiffeners on websis presented. The design is intended to be formulated based only on the actual version of Eurocode 3 Part 1-5Plated structural elements. Eurocode 3 will be definitely introduced in all states of the European Union. Theoptimization resources is an engineering compromise not only with the society, but also to avoid negativeimpacts on the environment. Typical box girder cross sections used in bridge design are shown in Figure 1 and Figure 2 without 1 INTRODUCTION and with longitudinal web stiffeners respectively 1.1 GeneralThe design of bridges is a major field of the engi-neering profession. The designer’s experience, intui-tion and ingenuity are required not only in the designof bridges. Optimization means the use of geneticalgorithms which imply the analysis of several trialsystems in a sequence before an acceptable design isobtained. It is therefore a challenge for an engineerto design efficient and cost effective bridges withoutcompromising their integrity and reliability. Theoptimal design process forces the designer to identi- Figure 1. Box girder cross section without stiffener.fy explicitly a set of design variables, a cost functionto be minimized, and the constraint functions for thebridge. Proper mathematical formulation of the de-sign problem is a key to good solutions. 1.2 Objective and scopeThe main aim of this study is to show how the rulesconsigned in a design code, for example, CEN(2005, 2006) Eurocode 3 Part 1-1 and 1-5 may pro-vide a rigorous mathematical formulation for an op-timization task. The second aim of this study is to check how all Figure 2. Box girder cross section with longitudinal stiffenerdesign rules of EC 3 Part 1-1 and Part 1-5 are articu-lated each to another in order to conduce a rationaldesign. If there are no criteria for the design in EC3another norm is used, but in accordance with theinspiration and philosophy of the Eurocodes.
L d all £ 2 DESIGN CODE 800 (12)Design provisions are taken from EC 3 Part 1-1 andPart 1-5. 3 OPTIMIZATION PROCEDURE Optimization means solving problems in which oneM Ed 2.1 Flexural strength £ 1,0 seeks to minimize or maximize a function by sys-M cRd (1) tematically choosing the values within an allowed Wel f yM c , Rd = set. gM0 (2) General search and optimization techniques are classified into three categories: enumerative, deter- ministic and randomized or stochastic.VEd 2.2 1 Shear strength leading to no M-interaction Enumerative techniques are the simplest tech- £ niques. Deterministic techniques incorporate moreVc ,Rd 2 fy (3) mathematical knowledge of the objective function.Vc , Rd = Av gM0 3 Stochastic techniques work with randomized values (4) as suggested solutions. In view of optimizing the problem of box girders 2.3 Local buckling considerations the following definitions are necessary:c/t ratios to avoid local buckling of the compressionflange, upper and lower part of the web, and hori- 3.1 Design variableszontal and vertical part of the stiffener are given in Design variables must be able to describe the designEquation 5 to Equation 9 respectively. problem. Following design variables are considered for this study:c / t £ 42e 42e (5)c/t £ 0,67 + 0,33y ì x1 = b f ü (6) ïx = t ïc / t £ 62e (1 - y ) - y ï 2 ï f ï ï (13) (7) x = í x3 = hw ý ïx = t ïc / t £ 33e (8) ï 4 w ï ï x5 = t s ï î þc / t £ 9e (9) 3.2 Lower and upper bound of design variables 2.4 Torsional buckling Design variables are limited through the lower and upper bounds as indicated in Equation 14.The stiffener in Figure 3 should meet the followingrequirement in order to avoid torsional buckling. IT fy x LB £ x £ xUB (14) ³ 5,3 IP E (10) 3.3 Design or objective function The objective function is a criterion needed to judge if the quality of a solution is better than another. In this study the objective function is the cross area: f ( x ) = 2( b f t f + hw t + b s t s ) (15)Figure 3. Detail of a typical longitudinal web stiffener. 3.4 Design constraints In case of2 non-fulfillment of Equation 10 DIN Constraints are functions which limit the interaction IT æ t ö18800 (1990) gives as alternative the Equation 11. between the design variables; they can be equalities ³ 11ç ÷ IP çb ÷ or inequalities as showed in Equations 16 and 17. è pø (11) h j ( x) = 0 (16) 2.5 Serviceability g k ( x) £ 0 (17)The allowed deflection is defined here as All geometric, design and usability conditions are transformed into constraints.
M Edg1 ( x ) = - 1,0 £ 0 Wy ( x) f y / g M 0 VEd 1 (18) 4.5 Mutationg 2 ( x) = - £0 A ( x) f y / g 3 2 This operator forms a new chromosome by making, b fv - 2t w M 0 (19)g 3 ( x) = - 42e £ 0 usually small, alterations to the values of genes in a tf copy of a single parent chromosome. h 1/ 5 42e (20)g 4 ( x) = w - £0 tw 0,67 + 0,33y upp h 4/5 (21) 4.6 Reproductiong 5 ( x) = w - 62e (1 - y low ) - y low £ 0 tw (22) Reproduction takes place throughout the two basic t Ig 6 ( x ) = 11 w - T £ 0 operators: crossover and mutation. d req w1) 5 I P h(x / (23)g 7 ( x) = - 1,0 £ 0 d all (24) 4.7 Next generation Next generations are not created until two or more solutions are close enough to each other or the max- 3.5 Formulation of an optimization problem imum number of generations is not reached.All mentioned variables, bounds, objective functionand constraints serve to formulate the optimization 4.8 Pseudo codetask in its canonical form: min f(x) xi Î RN i = 1,…N Table 1 shows a genetic algorithm outline.subject to hj (x) = 0 j = 1,… M (25) gk (x) £ 0 k = 1,… P Table 1. Pseudo code ______________________________________________and xLB £ x £ xUB Genetic Algorithm ______________________________________________ Begin Set maximal number of generations 4 GENETIC ALGORITHMS Create a randomized generation Repeat: Fitness evaluationThinking in terms of real life evolution helps to un- Selectionderstand genetic algorithms. The solution of N vari- Crossoverables is comparatively analogue to a chromosome. Mutation Reproduction: Create a new population Until Population size 4.1 Random Initial State End ______________________________________________An initial population of possible candidates to solu-tion is created at random. 5 NUMERICAL EXAMPLE 4.2 Fitness evaluation In order to validate the results of the optimizationValue for fitness is assigned to each candidate to the model presented by Jármai & Farkas (2001) andsolution (chromosome) depending on how close it is optimized by the “advanced backtrack method” isactually to solve the problem optimally. adopted. Additionally, a new constraint condition related to the usability is introduced, in order to prove the quality of the method. 4.3 Selection Structural data, geometry, material and general load-According to Holland (1975) and Goldberg (1989), ing are summarized in the Figure 4.natural selection means that some entities live andothers die according to fitness values. 4.4 CrossoverThe Crossover operator entails choosing two indi-viduals to swap segments of their parents. This pro-cess is intended to simulate the analogous process ofthe sexual reproduction. Figure 4. Structural system, material and loading. 6 RESULTS Tables 2,3 and 4 show the results of optimization.
Table 2. Solutions for a box girder without stiffener______________________________________________ application of the Equation 10 needs to be explainedMethod bf tf hw______________________________________________ tw in EC 3 Part 1-5, Equation 11 was used in this paper.Genetic 470 22 1122 9algorithm 540 19 1113 9 494 24 1033 9 8 REFERENCESBacktrack in 540 20 1100 9Járrmai (2001) 740 20 900 8 CEN 2005: European Committee for Standardization Eurocode 730 20 910______________________________________________ 8 3:Design of steel structures. Part 1-1. General rules andall dimensions in mm. rules for buildings. EN 1993-1-1. Brussels: CEN. CEN 2006: European Committee for Standardization EurocodeTable 3. Solutions for a box girder with stiffener ._____________________________________________ 3: 2006: Design of steel structures. Part 1-5. Plated struc-Method bf tf hw tw ts tural elements. EN 1993-1-5. Brussels: CEN._____________________________________________ DIN 1990.: Deutsches Institut für Normung e.V. DIN 18800Genetic 427 16 1596 7 5 Part 3. Berlin: Beuth Verlag GmbH.algorithm 473 15 1567 7 5 Goldberg, D.:1989. Genetic Algorithms in Search, Optimiza- 486 14 1602 7 4 tion, and Machine Learning. MA: Addison Wesley. 347 15 1774 8 6 Holland, J.H.: (1975) Adaptation in natural and artificial sys-Backtrack in 470 17 1440 6 5 tem. Massachusetts: MIT Press.Jármai (2001) 490 16 1440 6_____________________________________________ 5 Jármai, K & Farkas, J. 2001. Optimum cost design of weldedall dimensions in mm. box beams with longitudinal stiffeners using advanced backtrack method. Struct. Multidisc. Optim. (21) 52-59.Table 2 and Table 3 show that genetic algorithmsand Backtrack method give similar results. 9 NOTATION AND DEFINITIONSTable 4. Solutions for a box girder with stiffener and deflec-tion constraint g7(x) ._____________________________________________ bf Flange widthMethod bf tf hw_____________________________________________tw ts tf Thickness of flanges;Genetic 576 17 1943 9 6 hw Web heightalgorithm 513 19 1999 9 5 tw Web thickness 619 17 1989 9 6 ts Stiffener thickness 616 18 1946 8 6 bs Horizontal length of the stiffener_____________________________________________all dimensions in mm. hs Vertical length of the stiffener bp Shorter width of the web between stiffeners a Separation between diaphragmsTable 4 shows the results when a deflection con- MEd Design momentstraint g7(x) according to Equation 24 is introduced. Mc,Rd Resistance design momentThe reason in the new values lies in the fact that the VEd Design shear forceweb height should fulfill this new constraint too. Vc,Rd Resistance design shear force Wy Section modulus Av Shear area IT St. Venant torsional constant for the stiffener alone IP Polar second moment of area of the stiffener alone c/t Width to thickness ratio E 210.000,00 N/mm2 fy 235 N/mm2 ψupp Stress ratio for the upper part of the web ψlow Stress ratio for the lower part of the webFigure 5. Comparison between the possible solutions from γM0 1,00Table 2, Table 3 and Table 4. 235 f y ε = Material ratio δreq Required deflection 7 CONCLUSIONS δall Allowed deflectionGenetic algorithm is a promising tool to assist de-sign engineers in selecting optimal designs; theywork with possible solutions rather than determinis-tic ones. Design rules consigned in EC 3 Part 1-1 and 1-5,are capable of converting into constraint functionsfor an optimization of girder box cross-sections. Longitudinal stiffeners attached to a web of a boxgirder and subject to compression are special casesof the problem of torsional buckling. In this way the