Optimization of cable-stayed bridges with box-girder decks

Transcription

1 Avances in Engineering Software 31 (2000) Optimization of cable-staye briges with box-girer ecks L.M.C. Simões*, J.H.J.O. Negrão Department of Civil Engineering, University of Coimbra, 3049 Coimbra Coex, Portugal Abstract Box-girer ecks are very effective solutions for long span cable-staye briges, ue to their high torsional stiffness an streamline profile, which usually lea to a goo aeroynamic behaviour. A stuy on the optimization of such structural system is presente in this paper. The eck is moelle through the assembly of planes of plate-membrane elements. A multicriteria approach is consiere for the optimization itself, with constraints on maximum stresses, minimum stresses in stays an eflections uner ea loa conition. Two illustrative examples are shown Elsevier Science Lt. All rights reserve. Keywors: Cable-staye briges; Optimization; Box-girer ecks 1. Introuction Cable-staye briges are large an sophisticate structures which may greatly benefit from the use of structural optimization techniques for preliminary esign improvement. General optimization packages are not appropriate for this purpose given the special characteristics of this structural system an the rigorous aesthetical an serviceability esign. Therefore, an integrate analysis-optimization application for this type of structure was evelope by the authors, in which the require aaptations were implemente at the coe level. The programme was name cablestaye briges integrate analysis an optimization (CIAO). This programme was teste in the investigation of some relevant aspects that affect the esign of a cable-staye brige. The research moel on shape an sizing optimization of cable-staye briges starte by using a 2D finite element moel for the analysis [1]. The problem was extene to three-imensional analysis an consieration of erection stages uner static loaings [2,3]. Seismic effects must be consiere in structural esign in earthquake-prone countries. Therefore, a reliable solution requires the consieration of this loaing at the early stage of optimization. A sensitivity analysis algorithm erive from the moal-spectral approach in connection with the complete quaratic combination metho (CQC) [4] was later escribe [5]. However, in instances such as with strongly non-linear behaviour or when coalescence situations arise, the algorithm may not be vali or lea to runtime error. The alternative use of a time-history base sensitivity analysis * Corresponing author. proceure is then recommene, though computationally expensive. Such algorithm was also implemente in the software. Recent papers [6,7] iscuss the specific issues an relative merits of both methos. 2. Analysis moel 2.1. Deck In most of the previous stuies, a gri solution was aopte for moelling the eck, with sie stiffening girers supporting transverse beams. However, box-girer ecks provie increase torsional resistance an better aeroynamic behaviour than open sharp eck profiles, which are ecisive aspects in the esign of long-span cable-staye briges. Besies, when certain structural solutions such as the single plan arrangement of cables are aopte, the use of box-girer sections is more appropriate. This paper concerns the optimization of cable-staye briges with this type of eck. Prior to the coe implementation, one ha to choose the numerical moel to be use from among three possible types: (i) spine iscretization, in which the eck is moelle by a fictitious beam with global stiffness characteristics similar to those of the true box-girer cross section; (ii) specific box-girer element formulation; (iii) assembly of plate-membrane plane elements. The former approach was use in an early paper of the authors [1], although the two-imensional analysis unertaken in that stuy i not allow for the consieration of out-of-plane behaviour. However, in spite of being computationally afforable, this moel requires that /00/$ - see front matter 2000 Elsevier Science Lt. All rights reserve. PII: S (00)00003-X

2 418 L.M.C. Simões, J.H.J.O. Negrão / Avances in Engineering Software 31 (2000) pre- an post-processing is provie to set the fictitious beam imensions an to account for box-girer effects such as warping an shear-lag. Furthermore, as far as optimization is concerne, a two-level proceure must be use. Fictitious cross-sectional areas an inertias are use as intermeiate esign variables from which the optimal values of the primary ones plate thicknesses an cell imensions are in turn optimize. Specific box-girer elements inten to conciliate computational effectiveness an analysis accuracy. Nukulchai an Hong [8] formulate one such class of elements. However, a ifferent number of egrees of freeom is assigne to the element noes, which may require some major changes of the analysis coe. Besies, these elements are usually not available in the libraries of most f.e. packages an therefore its full evelopment is require. Finally, the elements lack flexibility for moelling ifferent cross-section geometries, such as multicell box-girers, sie cantilever plates an iaphragms. The latter approach was, therefore, selecte for use in the programme. Given its suitability for both thin or thick plate problems, Reissner Minlin formulation was consiere an 4- an 9-noe isoparametric lagrangean platemembrane elements were evelope. A selective quarature proceure was use for the stiffness matrix, with a 2 2 rule for the bening terms an a single point rule for shear an membrane sub-matrices. A fictitious in-plane rotational stiffness was ae to coplanar noes in orer to prevent system singularity. Null-energy moes may arise from the use of selective quarature, causing the solution process to fail. However, in the specific type of structure uner stuy, they were foun not to propagate through the mesh, ue to the intersection of box-girer element planes an the bounary conitions at the extreme cross-sections Stays Fig. 1. Sizing esign variables. Most cable-staye briges show moerate non-linear behaviour, which is contribute, from three main sources: the sag effect of the cables, the effect of large isplacements an P D interaction. Therefore, some provisions to account for non-linear effects nee to be consiere an the use of a geometrically non-linear analysis approach shoul look the obvious choice. However, although the two latter effects may play an important role in instances such as alternate loaing in large spans an/or erection stages, most authors agree in that the catenary effect of the cables is the ominant source of non-linearity. Besies, one is to expect very high computational costs an consierable numerical ifficulties in implementing sensitivity analysis for a pure geometrically non-linear approach. Therefore, the equivalent or Ernst moulus metho, which accounts for the non-linear behaviour of the stays while still allowing for the use of a pseuo-linear analysis, was chosen. The main problem concerning this metho is that ifferent equivalent moulus must be assigne to each cable for each loa case (accoring to its actual stress conition) when several loa conitions are consiere. This woul require the equation solver to be restarte for each loa case, which is extremely expensive in a single analysis process an even more in an optimization context. Fortunately, the variation of the equivalent moulus within the range of cable stresses spanning the various loa cases is usually small. In fact, large minimum tensile stresses are require to act in the stays, in orer to provie them with the aequate stiffness. Such type of conition can be ae as a constraint in the optimization formulation, to guarantee a minimum level of stress. Thus, an average equivalent instantaneous Ernst moulus was use, for each stay, for all the loa cases uner consieration. When erection stages are accounte for, the stress amplitue from one stage to another may be consierably large an the mean stress is usually smaller than average stress acting in the cable in the final structure. In such case, since the previous approach is no longer vali, a secant equivalent Ernst moulus whose value is erive from the stress in both the current an the following stage is use instea. However, this topic shall not be iscusse in etail in this paper. 3. Design variables Selecting a trial esign for a cable-staye brige involves setting a large number of parameters of both sizing, shape an mechanical (prestressing) types. Owing to the complex Fig. 2. Shape esign variables.

3 L.M.C. Simões, J.H.J.O. Negrão / Avances in Engineering Software 31 (2000) Sensitivity analysis 4.1. General Fig. 3. Fixe-en prestressing esign variable. structural behaviour an parameter interaction, it is not easy to foresee how to combine their values in orer to achieve a reasonably effective an feasible esign. For the optimization sake, it is esirable that a wie choice of potential esign variables is available. This was achieve by introucing the concept of esign variable library. Basically, it consists of a set of proceures an instructions on how to compute goals an sensitivities for a set of previously efine parameters. These concern geometrical escription for a number of cross-section shapes, overall structural geometry an cable prestresses. Figs. 1 3 show the relevant esign variable types for the kin of problem being iscusse. Other types are available for plate girer type ecks. Different thicknesses may be assigne to the various panels of the box-girer cross section. Shape cross-sectional esign variables such as box height or bottom with, which may play an important role in future research concerning seismic or aeroynamics problems, are currently uner evelopment. Maximum structural cost ecrease is achieve with formulations in which the three types of esign variables are combine. Sizing esign variables provie for irect volume/cost reuction. Shape an prestressing esign variables, with respect to which cost is almost insensitive, play nevertheless an important role in allowing for better stress istribution, leaing to aitional sizing reuctions. Prestressing esign variables may be consiere alone for the problem of optimal correction of cable forces uring erection [9]. Several trial esigns may be teste by simply switching on or off each type of esign variable or its assignment to specific subomains or groups of elements. This intensive ata process is automatically hanle by the programme. Pre- an post-processing routines are available for initial generation an upating of the f.e. mesh an variable linking when shape esign variables are consiere. However, care must be taken to avoi excessively narrow plate elements in the transverse irection, which causes egeneration of the numerical accuracy. This may happen because longituinal iscretization accounts for both the cable anchorage positions, which vary throughout the process, an the iaphragms spacing, which is likely to be a prescribe value. In such cases, a refine iscretization must be use for both irections, which results in the increase of the computational cost. Given the availability of the source coe, the iscrete nature of cable-staye brige structures an the large number of constraints (stresses an isplacements) uner control, the analytical iscrete irect metho was use for the sake of sensitivity analysis. The metho relies upon the ifferentiation of the finite element equilibrium equation set K u ˆ P K u ˆ Q pi in which Q P is the virtual pseuo-loaing corresponing to esign variable x i Plate-membrane element sensitivity analysis The coefficients of the f.e. equation system result from assembling the element contributions. Therefore, the same applies to their erivatives in Eq. (1), which requires sensitivity analysis to be evelope at the element level. Plate-membrane elements of the eck show explicit epenency only on esign variables types 1, 4, 5, 7, 8, 10 of Figs. 1 an 2. Type 1 is a sizing esign variable an the remaining types are shape esign variables. Sensitivitity analysis is basically ifferent for each of these groups an a brief escription shall be mae in the following. From the f.e. formulation of Reissner Minlin platemembrane elements, one can notice that the coefficients of the element stiffness matrix are of either of the forms Z Z Z F h N i x; y N j x; y x y V V 1 2a F h N i x; y N s j x; y x y s ˆ x or y 2b F h N i x; y N j x; y V s r x y s ˆ x or y; r ˆ x or y 2c in which h is the plate thickness, (x,y) are the element noe cartesian local co-orinates an N j are the element shape functions. For the purpose of systematic application, Eqs. (2a) (2c) are usually expresse in terms of intrinsic co-orinates, leaing to F h N i j; h N j j; h J j h F h f s j; h; x e N j j;h j h 3a s ˆ x or y 3b

4 420 L.M.C. Simões, J.H.J.O. Negrão / Avances in Engineering Software 31 (2000) F h f s j; h; x e f r j; h; x e 1 j h J s ˆ x or y r ˆ x or y where J is the Jacobian matrix an fk x j; h; x e ˆ 2N k 2h 2j 2N k 2j 2h for ifferentiation with respect to x f y k j;h;x e ˆ 2N k 2h 2j 2N k 2j 2h for ifferentiation with respect to y 3c 4a 4b The erivatives of the cartesian local co-orinates are erive from the isoparametric relations involving the element N noes co-orinates One has for the sensitivity of the Jacobian eterminant J 2j 2h 2h ˆ 2h 2j 2j 2j 8 2h By ifferentiation expressions for functions f in Eqs. (4a) an (4b): f x k 2h 2N ˆ k 2j 2j 2N k 2h f y 9 k 2h 2N ˆ k 2j 2j 2N k 2h x ˆ XN N j j; h x j y ˆ XN N j j; h y j 5 Derivatives of cartesian local co-orinates in Eqs. (8) an (9) are finally obtaine from ifferentiation of Eq. (6), resulting in: 2j ˆ XN 2h ˆ XN N j j x j N j h x j 2j ˆ XN 2h ˆ XN N j j y j N j h y j By ifferentiation of Eqs. (3a) (3c) with respect to the esign variable x i one can get now the sensitivities of the stiffness coefficients. For the case in which a sizing esign variable (plate thickness) is being consiere, only the function F(h) which sets strain variation through the plate epth leas non-zero erivative an the sensitivity shall be reaily available. When ealing with shape esign variables, all the remaining terms in Eqs. (3a) (3c) are variable epenant an the sensitivity expression results somewhat more complex: 6 F h N i j; h N j j; h J j h 7a F h f s j;h;x e N j j;h j h 7b " f s j; h; x F h e f r j; h; x x e i f s j; h; x e f r j; h; x e ˆ XN N j x j 2j j x i ˆ XN N j y j 2h h x i ˆ XN N j y j 2j j x i ˆ XN N j x j 2h h x i 10 The erivatives of the noal co-orinates with respect to the shape esign variables are, therefore, the basic brick to buil all the require sensitivities. The particular shape of the omain, with all box-girer plate elements boune by sies which are either parallel or orthogonal to the longituinal irection makes that an easy task. 5. Optimization 5.1. The algorithm NLP algorithms may face consierable ifficulty in ealing with problems in which hunres or thousans of constraints are to be controlle. Since the simultaneous reuction of all the objectives (stresses an isplacements) is esirable in structural optimization, one may employ a minimax approach, but these problems are iscontinuous an non-ifferentiable an therefore ifficult to solve. However, it may be shown [10,11] that a minimax solution can be foun by minimizing the unconstraine scalar function f r j; h; x e f s j; h; x e 1 J # J j h J 7c F x ˆ 1 XM ln e rg i x r iˆ1 11

5 L.M.C. Simões, J.H.J.O. Negrão / Avances in Engineering Software 31 (2000) Fig. 4. Geometry of starting esign. or, in the explicit approximate form 0 1 X F x ˆ 1 r@ N 2g g i x 0 i x 0 XM j Dx j A ln e r iˆ1 ˆ 1 XM ln e r g i x 0 7 T g i Dx 12 r iˆ1 which is more suitable for numerical calculation. N is the number of esign variables, g i (x) are the M objectives, fg(x) stans for objective graient an Dx is the perturbation vector. r is a user-efine parameter control which must be graually increase throughout the iterative sequence Constraints In orer to prevent numerical inaccuracy, the objectives are turne into non-imensional normalize form by using reference values which are: the volume/cost of the starting trial esign; the allowable stress for materials; the tolerance eflection for geometry control. Accoring to this, the following types of objectives are formulate: g 0 x ˆV=V 0 0 Cost reuction goal 13 g i x ˆs i =s A 0 g i x ˆa s i =s A 0; 0 a Maximum allowable stress 14 Minimum require tensile stress in stays for effective stiffness 15 g j x ˆ j = 0 0 Controlle eflections: Examples The process is illustrate by two examples that iffer in the consieration of geometry control constraints (16). The geometry of the starting esign is shown in Fig. 4. A relatively large span ha to be use in orer to achieve significant stresses on eck, since the lower boun of 15 mm, impose for the equivalent thickness of the steel plates, result in a consierably large cross-sectional area. A single cell box-girer with the size shown is use. Diaphragms are 10 m space, in orer to match all cable anchorage positions in both sie an central spans. Minimum stress in stays is prescribe as 20% of allowable stress. For controlling the geometry in the ea loa conition of problem 2, a tolerance of 5 cm is allowe for eflections at the eck anchorage points. Besies ea loa conition neee for geometry control, another four loa cases are consiere: live loa on sie an/or central spans (3) an lateral win (1). Dea loa is taken as 2.0 kn/m 2, aitionally to structural self-weight. For live loa a uniform loa of 4.0 kn/m 2 is consiere, while as for win a rag pressure of 1.5 kn/m 2 is use, no lifting forces being accounte for. Allowable stress is taken as 200 MPa for pylons, 140 MPa for eck plates an 500 MPa for stays. 47 esign variables were consiere in both problems, with the following meaning an sequence: 1, 2, 3 equivalent thicknesses (mm) of top, bottom an sie plates, respectively, of each sie span outer 160 m; 4, 5, 6 the same, for 30 m on each sie of the pylons; 7, 8, 9 the same, for remaining part of central span; length, with an thicknesses of pylons below the eck (m, m, mm) the same, for pylons above eck level cross-sectional areas of cables, starting with the back-stays (cm 2 ) corresponing fixe-en prestressing forces in the cables (MN) 46 length of en block zone of cables in pylons (m);

6 422 L.M.C. Simões, J.H.J.O. Negrão / Avances in Engineering Software 31 (2000) Fig. 5. Cost reuction (%) against number of iterations: (a) no eflection control; an (b) with eflection control. Fig. 6. Initial an final maximum stresses in stays an pylons. Fig. 7. Deflecte shape for ea loa conition in starting esign an solutions 1 (no geometry control) an 2 (with geometry control).

7 L.M.C. Simões, J.H.J.O. Negrão / Avances in Engineering Software 31 (2000) Table 1 Starting an final values of esign variables for examples 1 an 2 DV St. Ex1 Ex istance from eck level to the lowest cable en block (m). Fig. 5 represents the cost reuction for both solutions. The starting esign shows a constraint violation with respect to the minimum stress conition in the back-stays, when the live loa on the sie spans is acting on the structure. The optimization process looks for the esign feasibility in the first place an that results in an initial increase in cost, which is recovere later. When eflection control is consiere, the cost recovery takes place after a larger number of iterations. This is ue to the arisal of several other constraint violations relate to the eflection of controlle noes, aitionally to the minimum stress conition in the back-stays. As a result, the final cost reuction is just about 6.5% of the initial cost, against 10% for the example without eflection control. Fig. 6 shows the initial an final stress istribution in the pylons an stays for example 1, similar results being achieve for example 2. However, the kinematic behaviour is rather ifferent in each optimize esign, as one can observe in Fig. 7. The eflecte shape of problem 2 coul still be enhance if a smaller tolerance were prescribe for checking the controlle isplacements. The sag of spans between consecutive stays epens only on local flexural behaviour an must be prevente by properly cambering the eck segments in the fabrication site (Table 1). In both optimal esigns the final arrangement of stays approaches the fan configuration, the en-block zone length converging to the lower boun. Conversely, the istance from the eck to the lowest cable anchorage reaches the upper boun, which provies the stays with maximum slope an vertical stiffness. References [1] Simões LMC, Negrão JH. Sizing an geometry optimization of cablestaye briges. Computers an Structures 1994;52: [2] Negrão JH, Simões LMC. Optimization of cable-staye briges with three-imensional moelling. Computers an Structures 1997;64: [3] Negrão JH, Simões, LMC. Three imensional nonlinear optimization of cable-staye briges. In: Topping B, Paparakakis M, eitors. CST/94 avances in structural optimization. Proceeings of the Secon International Conference on Computational Structures Technology, Athens, Greece, Civil-Comp, Einburgh, p [4] Wilson EL, Kiureghian A, Bayo EP. A replacement for the SRSS metho in seismic analysis. Earthquake Engineering an Structural Dynamics 1981;9: [5] Simões LMC, Negrão JH. Optimization of cable-staye briges subjecte to ynamic loaing. In: Olhoff N, Rozvany G, eitors. WCSMO1. Proceeings of the First Worl Congress of Structural an Multiisciplinary Optimization, Goslar, Germany, Pergamon, Oxfor, p [6] Negrão JH, Simões LMC. Effectiveness of moal/spectral versus timehistory methos. In: IUTAM, e. Optimization of cable-staye briges uner seismic action, in 19th ICTAM. 19th International Congress on Theoretical an Applie Mechanics, Kyoto, Japan, p.268. [7] Simões LMC, Negrão JH. Comparison between moal an step-bystep approaches in the optimization of cable-staye briges subjecte to seismic loas. In: Soc. Mex. Ing. Sismica, e. 11WCEE, paper/cd- ROM. Proceeings of the 11th Worl Conference on Earthquake Engineering, Acapulco, Mexico, Pergamon, UK, [8] Kanok-Nukulchai W, Hong G. Nonlinear moelling of cable-staye briges. Constructional Steel Research 1993;26: [9] Negrão JH, Simões LMC. Cable stretching force optimization in cable-staye briges. In: Mróz Z, eitor. WCSMO2. Proceeings of the Secon Worl Congress of Structural an Multiisciplinary Optimization, Zakopane, Polan, [10] Simões LMC. Information entropy applications in structural optimization. In: Hernanez S, Brebbia CA, eitors. OPTI 91. Optimization of structural systems an inustrial applications, Cambrige, MA, Elsevier, p [11] Templeman AB. Entropy-base optimization methos for engineering esign. In: Hernánez S, eitor. Avance techniques in the optimum esign of structures, topics in engineering, vol. 12, Computational Mechanics Publications, UK, p