Finite Strip Modeling for Optimal Design of Folded Plate Structures

1054 A. Bergamini, F. Biondini / Engineering Structures 26 (2004) 1043–1054adopted to check the convergence of the iterativeprocedure. Finally, Fig. 14 shows the correspondingevolution of the objective function f(x) andof its componentsf V (x) and f P (x).4.5. Box-girder bridge deck with internal diaphragmsThe box-girder bridge deck shown in Fig. 15(a), withlength L ¼ 40:0 m, total width B ¼ 16:0 m andheightH ¼ 2:0 m, is considered. Besides its self-weight, theweight of the non-structural elements g p ¼ 4kN=m 2andthe service loadp ¼ 2:0 kN=m 2 , the structure issubjected to the four alternative loading conditionsshown in Fig. 15(a), with q ¼ 8kN=m 2 . The thicknessdistribution andboth the shape andtopology of thecross-section which minimize the total volume, are searchedfor.To this purpose, the topologies associatedtothe four alternative symmetrical cross-sections shownin Fig. 15(b), each of them derived from the first fundamentalone by eliminating or not one or more internaldiaphragms, are considered. The search processleads to an optimal solution which corresponds to thetwo-cellular box girder shown in Fig. 15(c), with a totalvolume Vðx opt Þ¼7:737 m 3 =m. It is worth noting thatthe high value of thickness of the bottom slab intension is due to the hypothesis of homogeneousmaterial andelastic behavior. The disposition ofproper longitudinal reinforcement, designed with referenceto the effective composite andnon-linear nature ofthe material, clearly allows a lower thickness of theconcrete slab.Finally, with reference to the path of the search process,Fig. 16 shows the distribution of the topologies atthe vertices of the complex andthe evolution of theobjective function f(x) for the current best solutionduring the iterative process.5. ConclusionsThe problem of finding the optimal shape, size andtopology of prestressed folded plate structures undermultiple loading conditions has been investigated. Thedesign problem has been formulated as a mathematicaloptimization problem, accounting for both static andkinematic constraints, andit has been solvedby usinga numerical algorithm basedon the complex method.The structural analyses needed for the optimizationprocess have been performedby using the finite stripmethod. A number of applications have shown theeffectiveness of the proposedprocedure.In particular, the design tools presented in this papercan usefully be appliedat the conceptual design stage,where the designer is usually interested in comparingbetween them a variety of alternative optimal solutions,derived, for example, by using different designmodels. These solutions, which the proposed procedureidentifies with wide generality, rationality and objectivity,can be then usedas a basis for a more detaileddesign of the reinforcement. Clearly, future developmentsare expectedin order to obtain a better controlof the design problem, especially on the definition ofthe design constraints, which should also account fortechnological aspects, as well as for aesthetical andadditional functional requirements.References[1] ASCE. Design of cylindrical concrete shell roofs. Manual ofengineering practice. New York (NY): ASCE; 1952, p. 31.[2] Belegundu AD, Chandrupatla TR. Optimization concepts andapplications in engineering. Upper Saddle River (NJ): Prentice-Hall; 1999.[3] Bergamini A, Biondini F. Optimisation of folded plate structures.Proceedings of the Second International Conference onAdvances in Structural Engineering and Mechanics (ASEM02),Seoul, Korea, August 21–23. 2002.[4] Biondini F, Bontempi F, Malerba PG. Ottimizzazione di formadi strutture a folded-plates. Proceedings of the 13th C.T.E. Conference,Pisa, Italy, November 9–10–11. 2000 [in Italian].[5] Cheung YK. Finite strip methodin structural analysis. Oxford:Pergamon Press; 1976.[6] Kristek V. Theory of box girder bridges. Prague: John WileyandSons; 1979.[7] Loo YC, Cusens A. The finite strip methodin bridgeengineering. London: E. & F.N. Spon; 1978.[8] Malerba PG, Toniolo G. Metodi di discretizzazionedell’analisi strutturale. Milan, Italy: Masson Editore; 1981[in Italian].[9] Martinez Y Cabrera F, Menni C. I ponti a cassone monocellularea profilo deformabile. Technical Report, 33. Istituto diScienza e Tecnica delle Costruzioni, Politecnico di Milano,Tamburini Editore, Milan, 1974 [in Italian].[10] Mortenson ME. Mathematics for computer graphicsapplications. New York (NY): Industrial Press; 1999.[11] Nielsen MP. Limit analysis andconcrete plasticity. Boca Raton(FL): CRC Press; 1999.[12] Rao SS. Engineering optimization—theory andpractice. NewYork (NY): John Wiley andSons; 1996.[13] Schlaich J, Scheef H. Concrete box-girder bridges. Structuralengineering documents, 1e. International Association for BridgeandStructural Engineering (IABSE); 1982.