Keywords: methodology, laminated structures, manufacturing tolerances, optimal design, minimum weight.

A methodology to design symmetrically laminated structures under transverse loads for minimum mass with manufacturing uncertainty in the ply angle, is described. The ply angle and the ply thickness are the design variables, and the Tsai-Wu failure criteria is the design constraint used. The finite element method, based on Mindlin plate and shell theory, is implemented, and thus effects like bending-twisting coupling are accounted for. In order to demonstrate the procedure, laminated plates with varying aspect ratio and boundary conditions are optimally designed and compared.

Optimal Design Problem and Solution Procedure

The objective of the design problem is to minimise the mass of the plate, with manufacturing uncertainty in the layup angle accounted for. The problem can thus be stated as

where the mass of a plate is given by and where is the total thickness of the plate. In this case the minimum mass is found by determining at each value of until (and thus ) is obtained, where is the value of the Tsai-Wu failure criteria at the value .

When composite laminates are manufactured, the desired fibre orientation in different plies may deviate from their intended design values by a few degrees [1,2]. These deviations are referred to as manufacturing tolerances. Assume that for the interval , a manufacturing tolerance in the layup angle is incurred. For example, there may be a maximum variation band of or , with In order to illustrate the problem, consider the following scenario: assume a manufacturer incurs the following maximum tolerances, and ; viz. when trying to achieve the value , the actual value that results is . Figure 1 shows the effect of the tolerance on the minimum required plate thicknesses for a (CCCC) laminated plate with four symmetric layers of equal thickness and with . There are three trendlines given, and these represent the nominal layer thickness along with the upper and lower bands. The design problem becomes one of determining the value of at which the layer thickness is minimized thus reducing the weight of the laminate, with the tolerances accounted for, which effectively becomes one of determining the value of for which the trend described by the upper composite line in Figure 178.1 is minimised. The effect of manufacturing tolerances on the optimal lay-up orientation for this plate is 51.43, which corresponds to a plate thickness of 10.89 mm. The optimal values for the nominal case are 48.84 and 10.10 mm. It is apparent from this example that the values of the actual optimal results are different from those of the nominal, and that if we were to ignore the manufacturing tolerances and choose 48.84 as the optimal fibre orientation, the corresponding minimum thickness required could be as high as 11.32mm (viz. the value at 48.84 - 7), which is 12% more than the optimal value.

The optimization procedure thus involves the stages of determining the minimum layer thickness required for a given , and to satisfy the constraint, selecting the greatest of the three values, and improving the fibre orientation to minimise the greatest value. Thus, the computational solution consists of successive stages of analysis and optimization until convergence is obtained and the optimal angle and layer thickness is determined within specified accuracy.

1

M. Walker, R. Hamilton, "A technique for optimally designing fibre-reinforced laminated plates with manufacturing uncertainties for maximum buckling strength", Eng. Opt, 37(2), 135-144, 2005. doi:10.1080/03052150412331298371

2

M. Walker, R. Hamilton, "A methodology for optimally designing fibre-reinforced laminated structures with design variable tolerances for maximum buckling strength", Thin Walled Structures (In Print). doi:10.1016/j.tws.2004.07.001

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