In this paper a simple new iterative method of altering the thickness profile of plate structures, called Plate Thickness Refinement (PTR), is presented. The method averages the design value contributions from those finite elements that share a common vertex edge and rescales the thickness of that edge in proportion with the ratio of the design value average to the target design value for the current analysis. The design information is stored within the finite element model and no classical optimisation process is carried out. Once the finite element mesh has been created, the finite elements in those sections of the structure specified, due to functional requirements, as being non-design areas, are not altered by the PTR program.

For ease of manipulation the input data is organised into three main structures, evolution control information, element information and nodal information. First the evolution control information is read which includes whether a target design value or optimal design value is sought and what are, if any, the maximum and minimum depth limits. Next, the stress contribution from each finite element for each node is calculated and the finite element strain energy contributions determined. Second the global reference datum for the current iteration is set as either a fixed target value or as a series of design values within the design range. For any specific design value a stable profile is achieved when further refinements result in no net change of material. Third a list of all node pairs which form element vertex edges are compiled. An average design value is then calculated over all elemental design value contributions from both nodes forming the vertex edge. This average is divided by the reference datum to provide a scaling index for that edge. New edge depths are then calculated by factoring the existing edge depths by their respective scaling indices. Fourth, once all new depths have been calculated they are checked against the allowable depth range. If any new depth exceeds these limits the new depth is reset to the depth limit. Lastly, the net change in material volume is then calculated and all convergence criteria checked. The itterations contiue until the target or optimal design value is achieved. The algorithm assumes a continuous thickness profile over the structure.

In two previous papers three case studies were used to demonstrate the effectiveness of the PTR method for ammending thicknesses in plate structures such that the von Mises stress distribution within, or stiffness of, the plate is altered towards a desired maximum or minimum [1,2].

In this paper the PDR method is more fully described and a new case study presented. The case study is a plate in the form of a deep cantilevered beam, with a point load in the middle of its free edge. The objective is to find a stiffer and materially efficient plate depth geometry for the cantilevered beam, not by minimising deflection, but by minimising the strain energy.

After the first iteration of the PDR algorithm, the finite element thicknesses had reduced and the maximum deflection almost doubled. Between iterations 2 to 7, the maximum deflection decreased more rapidly than the increase in material volume. Between iterations 7 to 12, both the rate of deflection and the increase in material volume had decreased. Between iterations 12 to 19, there was almost no change in material volume, strain energy or maximum deflection. Between iterations 19 to 31 the structure had regained its original volume and the maximum deflection was 42% of its original value. Diagrammatic output showed that the Von-Mises stresses had been reduced by redistributing material volume towards the top and bottom flanges, around the point load and diagonally between the point load and the flanges. These increased volumes were truss like in form. Between iterations 31 to 50 the material volume increased to 150% of its original value with the maximum deflection decreased by a further 30%. As there was no associated reduction in either strain energy or von Mises stresses the refinement was halted.

In essence, the results showed that the original maximum deflection could be achieved with 42% less material. Further, by rearranging the original volume of material, the maximum deflection could be reduced by 35%. After 52 iterations, the maximum strain energy was reduced by 97% and maximum deflection by 72%. However, 150% more material was required. Any design between the 3rd and 32nd iteration would have provide a stiffer structure, used less material, and have had a smaller maximum deflection.

The PTR algorithm has been applied successfully to a case study. PTR is a robust method applicable to a wide range of problems. PTR incorporates a capacity for functional requirements and requires no specific mathematical programming technique to be developed for individual problems.

1

Christie, W.C., Bull, J.W., "Optimizing thickness profiling of plate structures", submitted to "Third M.I.T Conf on Computational Fluid and Solid Mechanics", USA, Bathe K.J. (Editor), Elsevier, Oxford, 2005.

2

Christie, W.C., Bettess, P., Bull, J.W., "Through depth plate profiling for reduced stress concentrations of optimal stiffness", "First ASMO UK / ISSMO Conference on Engineering Design Optimization", Toropov, V.V. (Editor), MCB University press, Bradford, 111-118, 1999.

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