Keywords: cross section design, optimisation, spreadsheet analysis, biaxial bending, composite sections, spreadsheet modelling.

A method is presented to design and optimise arbitrarily shaped composite sections subject to biaxial bending and axial forces. It allows for the assumption of all kinds of linear or non-linear material behaviour and uses optimisation methods and numerical integration to solve for necessary strengthenings (such as the required amounts of bar reinforcement in a reinforced concrete section) or optimised section shapes. The method is very suitable for practical engineering applications, as it, on the one hand, is easy to implement into common spreadsheet analysis programs [1] and requires only negligible computing times and, on the other hand, provides precise solutions of complex design and optimisation problems.

The set of equations for the fundamental optimisation problem is derived from the standard form of optimisation problems and the equilibrium conditions of a slender spatial bar. In detail it consists of the following equations:

• The equilibrium conditions between the actions of the bending moments as well as axial forces and the corresponding resistances that arise from normal stress integrals over the cross section (equality constraints). Thereby, the Bernoulli hypothesis is assumed (plane sections remain plane) and separate linear or non-linear stress-strain relations describe the material behaviour for each particular material type included in a composite section.
• Inequality constraints limiting the extent of stresses, strains, dimensions, areas or variable material parameters
• A scalar objective function that minimises the total cross section area in the manner of cost or mass minimisations. The partitions of each material type are thereby weighed by specific factors.

Implementation and solution of the optimisation problem with the help of only a spreadsheet analysis program have two major aspects. First, the problem itself must be numerically solved. For this purpose, a Generalized Reduced Gradient algorithm (GRG) [2] as well as Newton methods are available in common programs like MS-Excel [3]. The second and even more important aspect is the necessary integration of stresses factored with distances over the cross section during each step of the iterative solution. This is also done numerically. Similar to the familiar discretisation into finite elements, the cross section is subdivided into rectangular area elements. The properties of each element, like its incremental area, medium y- and z-distances from the centroid, strain or stress, are attributed to cells in the spreadsheet. Starting from one cell the combining equations are effectively and illustratively transferred on large cell ranges using graphical "drag and drop" procedures. Moreover, shape and grid structure of such cell ranges (e. g. the range containing the strain values) correspond to the ones of the cross section itself. Thus, the implementation can be checked graphically. In a last step, stresses, area increments and distances are combined to increments of the axial force and of the two bending moments and summed up to the resultant resistances. Normally, simple integration rules are sufficient for materials like concrete, steel, wood or soil. The method is applied to one design and one shape optimisation example.

A reinforced concrete section with a double-cross shape should be designed against the action of a bending moment of increasing value. Reinforcement can be provided at three fixed points in the section and the non-linear material equations according to the ultimate-limit-state design methods in Eurocode 2 are used for concrete and steel. The equations of the optimisation problem are presented as well as figures of the spreadsheet implementation and graphics of the relations between bending moment and computed minimum amounts of reinforcement.

An eccentrically loaded rectangular footing with a pronounced gaping joint is optimised in its size, shape and position. Surrounding footings and soil stress limitations restrict the problem. Compressive soil stresses are assumed to be linearly distributed, while tensile stresses are excluded. Figures for the geometries and stress distributions in the initial and optimised state, extracts of the spreadsheet implementation as well as a graphic showing the developments of the optimisation variables during the iterative solution illustrate the successful optimisation. A reduction in the required foundation area of about 50% is achieved, whereas the maximum soil pressures stay almost constant.

1

S.G. Powell, K.R. Baker, "The Art of Modeling With Spreadsheets", Wiley, Hoboken (USA), 2004.

2

L.S. Lasdon, A. Waren, A. Jain, M. Ratner, "Design and Testing of a Generalized Reduced Gradient Code for Nonlinear Programming", ACM Transactions on Mathematical Software 4, 34-50, 1978. doi:10.1145/355769.355773

3

D. Fylstra, L. Lasdon, J. Watson, A. Waren, "Design and Use of the Microsoft Excel Solver", Interfaces 28, 29-55, 1998. doi:10.1287/inte.28.5.29

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