Thickness Selection For The Flanges

The section is treated as thin-walled so that the top and bottom flange thicknesses and the web thickness are small compared with the overall beam width and depth. These overall dimensions are prescribed by the user, and the program provides efficient thicknesses for the flanges and web, for given bending moments and shear forces applied in the plane of the web, and for a specified flexural rigidity.

The sketch shows the various layers in the cross-section. It is noted that the laminates that make up the beam are all of the same unidirectional composite plies layed-up in 0 and/or +45 and 45 layers. It can be seen that the connection between the web and flange is made by continuing the web layers around to the inner faces of the flanges. Continuous 45 layers are added to the outer flange faces as shown. These layers provide transverse strength to the flanges, and prevent the separated web layers from splitting at the top and bottom of the web. Because the web lay-up is constrained to be symmetric, this arrangement means that two of the flanges will have a symmetrical lay-up while the other two will be antisymmetric. It is, however, assumed in the analysis that all the flanges are symmetric and it is anticipated that any coupling terms that arise in the constitutive equations for the antisymmetric flanges will be small and can be neglected.

The loading on the beam is assumed to consist of given combinations of bending moment and shear force at the critical section of the beam along its span. The user provides values for the moments and shear forces for up to ten different load cases and the computer program calculates flange and web thicknesses for given overall beam section width and depth.

The analysis to obtain the layer thicknesses for the flanges and web of the I-section is controlled by a set of constraints that finally determine the detailed solution for the various stacking sequences. The constraints are as follows.

Allowable unidirectional layer strains in tension, compression and shear. 
Buckling constraints for flanges and web. 
Satisfaction of a minimum stipulated value for the overall flexural rigidity of the beam. 

The user is required to provide the overall beam dimensions and loading, and the mechanical properties for a single unidirectional fibre-reinforced material. With this information a rough estimate for the flange and web thicknesses is obtained by assuming a uniform shear load in the web and uniform end-loads in the flanges. The procedure is known as initial sizing, and takes into account web and flange buckling and fibre failure. At this stage the top and bottom flanges have the same thickness.

Having obtained a rough first estimate for the design, the various thicknesses are then adjusted using a more precise direct and shear load distribution across the section and more precise equations for flange and web buckling. The ensuing reserve factors are utilized to adapt thicknesses to satisfy the constraints. This re-design procedure thickens or thins layers according to certain rules that are based on expectations of how the layers will best reinforce the beam. These are referred to as heuristic design rules and allow for failure due to excessive strains, flange and web buckling, and low flexural rigidity.

The design procedure is iterative, and is terminated when the minimum reserve factor is between 0.99 and 1.05 inclusive, indicating that any one layer just reaches one of the allowable strain values of the material, or that buckling is about to occur, or that the minimum required overall flexural rigidity of the beam is about to be reached. No other optimisation is performed and it is not possible to ensure that all of the above conditions are met at the same time because of the complicated interaction between them when the thicknesses are changed. However, the procedure guarantees that the associated reserve factors are all greater than or equal to 0.99 and the final design will generally be an efficient solution to the problem.

The analysis provides two sets of thicknesses, one in which the top and bottom flange thicknesses are constrained to be equal, denoted Configuration 1, and the other where these thicknesses are determined independently and may be unequal, denoted Configuration 2.

For all runs the program outputs a standard header which is followed by an echo of the input data. The quantity of additional information provided in the output for successful computations is determined by the value assigned to the output specifier in the input.

For brief output, the echo of the input data is followed by a table which presents all relevant results for the design. The results include the thicknesses, number of layers, the final cross-sectional areas, together with the minimum reserve factors. This is supplemented with the number of iterations in which the computations reached a converged solution, or not, an abbreviated description of the failure mode and the load case number for which the minimum reserve factor was obtained.

For extensive output, the brief output described above is preceded by a listing of the initial design thicknesses and tables that contain detailed data generated by the program for each iteration step for both Configurations 1 and 2. The tables present the evolution of the thicknesses of the flange and web and, separately, the associated values of the reserve factors.