finite strip method in the analysis of concrete box bridges

8/8/2019 Finite Strip Method in the Analysis of Concrete Box Bridges

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Proc. Instn Civ. Engrs, Part 2, 1975, 59, Mar., 189-192

7679 DISCUSSION

Applications of the finite strip method inthe analysis of concrete box bridges

A. R. CUSENS & Y. C. LOO

Dr K. R.Moffatt and Dr P. T. K . Lim, Structures and Computers LtdThe finite strip method enables a three-dimensional structure (such as a box girder)to be treated as a two-dimensional problem, or a two-dimensional structure (such as aplate) to be treated as a one-dimensional problem. Such a reduction in the size ofthe problem, and hence in the amount of computational effort, is generally possibleonly in situations where the geometrical and material properties of the structure donot vary along one co-ordinate direction.39. The Authors are therefore to be complimented on extending the finite stripmethod to situations where the regular lengthwise geometry of a structure such as abox girder bridge is disrupted by the presence of intermediate supports and dia-phragms. However, the flexibility approach used by the Authors involves, in addi-tion o he standard finite stripprocedure, the solution of a set of n flexibilityequations, where n is the number of redundant force components or sets of self-equilibrating force components associated with the intermediate supports and dia-phragms. The flexibility matrix so formed is not banded and for large values of n(as in the case of some steel box girder bridges) the advantages in economy of thefinite strip method over the more general finite element method would be eroded tosome extent.40. We have used the finite element method for the analysis of multi-cell bridgedecks using a two-dimensional idealization involving line elements degeneratedfrom ordinary finite element^;^^*^^ a similar technique has been employed by Cris-field.21 Although the technique is strictly limited to structures with a horizontalplane of symmetry, it can be used for structures having arbitrary variations in geo-

metry, boundary conditions and material properties. (A combination of this tech-nique with the finite strip method should enable the three-dimensional problem ofmulti-cell bridge decks to be treated as a one-dimensional problem, with an accom-panying drastic reduction in computer cost.)41. To illustrate the accuracy and efficiency of this technique we analysed themodel described in 5 16 for the point load over an outer web, using a general purposefinite element programlg which incorporates these special line elements. Owingto symmetry, only half of the model was analysed with the idealization shown inFig. 20. The elements used for this particular analysis had the nodal degrees offreedom U, , W, ,, Ou, and E, . Fig. 21 shows that the theoretical and experimentalresults of deflexion and strains are in close agreement. For this problem, the data

coding took less than 30 minutes; the computer cost at commercial rates was approxi-mately f2 .Paper published: Proc. Instn Ciu.Engrs, Part 2, 1974, 57, June, 251-273.

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D I S C U S S I O N3 a t 59.33

= l78 _I_ 5 at 178 = 8901

L?

L?14lN-x .Position ofpoint load

Fig. 20. Finite element idealization of ix-cell bridge model (dimensions in mm)

42. A finite element or finite strip analysis, based on classical thin plate theory,will predict that in the region of a concentrated load the bending moments tend toinfinity (in accordance with the thin plate theory) as the analysis is refined. There-fore, the question arises as to what level of refinement of the analysis will give satis-factory results.43. We considered this problem in respect of the finite element method of analy-sis'' and presented tables, relating element size to patch load ize, by means of whicha mesh can be selected to give the correct local bending moments for a particular sizeof patch load. It might be expected that, ina similar way, the width of strips and thenumber of harmonics used in a finite strip analysis could be related to the size of apatch load.44. The Authors, however, have selected a 'convenient' transverse mesh size, andthen found the number of harmonics required to give convergence (see 3 20). Sucha procedure seems rather arbitrary in thatt has led only to practical guidelines on thenumber of harmonics required for an analysis. Could the Authors give a generalbasis for the selection of the strip width in theregion of a concentrated load?Professor Cusens and Dr Loo'In general we agree with Dr Moffatt and Dr Lim that the solution time required bythe extended finite strip procedure for a multispan bridge with intermediate stiffeningis higher than for a simply supported structure. However, whereas the matrix equa-tion (4) which results from the basic finite strip procedure must be solved repeatedlyaccording to the specified number of harmonics, the flexibility equation ( 5 ) needs tobe solved only once. Furthermore, it has been found in analysing actual multispanconcrete and steel box bridges that satisfactory simulations of intermediate supportsand deformable diaphragms generally involve the solution of fewer than sixty redun-dant forces. In such cases the extended finite strip procedure remains highly com-petitive with other methods of analysis.

* Now Department of Civil Engineering, Universityof Wollongong,New South Wales.190


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- inite elemento ExperimentalFig. 21. Distribution of deflexions and flange strains at mid-span of six-cell bridgemodel under point load at mid-edge

46. Dr Moffatt and Dr Lim describe the use of a two-dimensional finite elementidealization in the analysis of multi-cell box structures. Such a simplified approachmay not always produce acceptable bottom flange bending moments, especially whenconcentrated loads are notapplied directly on top of webs. Moreover the necessaryassumption of equal top and bottom lange thicknesses is a definite limitation to thisapproach.47. The accurate determination of local bending moments at points immediatelyunder isolated concentrated loads has always been a problem for engineers usingdiscretization methods based on thin plate theory. Dr Moffatt and Dr Lim havepresented recommendations for the choice of appropriate finite element meshes as aresult of an analytical study. We felt that in deriving practical guidelines for the

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D I S C U S S I O Neffective use of the finite strip method it was lnore realistic to adopt a semi-empiricala p p r ~ a c h . ~48. Investigation of various finite strip simulations for the asbestos cement beamand slab modela3 has shownhat, apart from theeed for a narrow strip immediatelyto the eft and to the ight of the concentrated load, the widths of all other strips haveno effect on the convergence of local bending moment values. The widths of thesetwo strips which have their common boundary at the point of application of theconcentrated load should be two to three times the thickness of the plate. In thesimulation given in Fig. 7(a) the use of small strips in regions remote from the loadwas convenient for standard datapreparation for the numerous load cases examined.49. With the provision of two narrow strips in the immediate proximity of anyconcentrated load, the onvergence of the analysis is dependent only on the summationof harmonics and the totals recommended in the Paper may be adopted for generaluse.References19. LIMP. T. K. and MOFFATTK. R. Generalpurpose finite element program.Proceedings of the symposium on bridge programreview. P.T.R.C., London,1971.20. LIM P. T. K. Elastic analysis of bridge structures b y the finite element m ethod.PhD thesis, University of London, 1971.21. CRISFIELD . A. Finite element methods for the analysis of multicellular struc-tures. Proc. Instn Civ . Engrs, 1971, 48, Mar., 413-437.22. LIMP. T. K. and MOFFATT . R. Finite element analysis of curved slab bridgeswith special reference to local stresses. In Developments in bridge design andconstruction. Rockey K. C. e ta l. (eds). CrosbyLockwood, London, 1972.23. Loo Y .C. Developm ents and applications of the finite strip method in the analysisof right bridge decks. PhD thesis, University of Dundee, 1971.

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finite strip method in the analysis of concrete box bridges

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