5.1. T-Inserts
The discontinuity of fibres in the centreboard-foil interface was solved by designing T-inserts within the foam structure as shown in
Figure 1a. The inserts were designed using unidirectional plies wrapped over a delta fillet with fibres running through the thickness, see
Figure 1b. In order to create a conservative design, it was proposed that the T-inserts should be independently capable of carrying the applied loads induced on the centreboard and foil. As the face sheets are also required to carry the loads a safety factor of 1.2 is used on the T-inserts.
Handbook calculations based on Euler-Bernoulli beam theory (see e.g., [
5]) were conducted, considering the resulting surge and sway loads on the centreboard and the resulting heave and surge loads on the foil, to obtain a first estimate of thicknesses t
1 and t
2 (
Figure 1b). In these preliminary studies, a simple maximum stress criterion [
6] was used, considering only compressive fibre failure (since that is lower than the tensile strength and compressive stresses are higher than the tensile) to estimate the amount of reinforcements needed in the T-inserts.
The T-inserts were then, in a second step, analysed by the finite element (FE) method to get the actual stress distribution and design safety factor. For this purpose, the composite plies on the outside where modelled as layered thick shell elements (the shell has various properties through its thickness to represent the different layers) which were node-by-node connected to solid elements utilised to model the delta fillet. The upper part of the T-insert was fully constrained and the foiling forces were applied to the bottom of the insert. To assess the risk of failure in any ply (initial ply failure), several failure criteria were simultaneously assessed (Maximum stress, Tsai-Hill and Hashin, see [
6]). This in order to obtain a conservative design approach were only the most critical criterion, providing the lowest safety factor against initial ply failure, is considered in any point in a laminate.
5.2. Centreboard with Integrated Centre Foil
Initially, handbook calculations were used to propose an initial design for the centreboard and the centre foil. First, the maximum shear forces were determined from equilibrium of forces. With this information available, the shear stress distribution over each cross section could be calculated [
5]. In order to do so, the cross-sections of the components were approximated as ellipses (compared to the true NACA and Wortmann profiles [
1]). The maximum shear stress was then compared against the shear strength of the core to make sure that the sandwich design was strong enough.
To design the CFRP face sheets, the maximum bending moments and axial forces were determined in each component using Euler-Bernoulli theory [
5] and considering each component as an idealised beam. With these moments and forces available, the required number of plies with fibres along the main loading direction, the longitudinal direction as indicated in
Figure 2, could be evaluated. As for the T-inserts, the maximum stress criterion for compressive fibre failure was evaluated under the conservative assumption that these plies should be able to carry the entire load. However, in order to account for unforeseen transverse and shear loads and to provide a robust design, plies with 45° and 90° fibre orientations were added in the final design.
In the subsequent FE analysis of the centreboard and the centre foils, both components were modelled with solid elements for the core and layered shell elements (cf. above) for the face sheets. Furthermore, the connection between the centreboard and hull was not considered explicitly. Instead, the centreboard top was treated as a completely fixed boundary. This provided accurate results at the foil—centreboard joint, which is a critical area of interest. For the failure risk assessment, the same approach as described above for the T-inserts was used.
5.3. Centreboard Casing and Hull Reinforcement
Under normal (non-foiling) conditions the vertical forces on the centreboard are small. However, when foiling, significant vertical forces from the foil must be transmitted through the centreboard to the hull. This means that, with the increased load from the foil, the attachment of the centreboard must be modified. Furthermore, the centre foil and the centreboard are an integrated component. Thus, it is no longer possible to use the conventional solution where the centreboard is inserted from above through the casing. Instead, the centreboard has to be introduced into the casing from below the hull and is kept in place by the lift force when the dinghy is in the water. As a consequence, the geometrical design of the casing was given the same shape as the upper part of the centreboard (to fully enclose the centreboard) with a lid secured on top. To reinforce the casing and transfer the applied loads to the existing hull, transverse supporting bulkheads had to be designed to be placed on each side of the casing. These transverse bulkheads must also provide the required 50% increase of the torsional stiffness of the hull.
All in all, three concepts for the casing and the hull reinforcements were developed and analysed with respect to stiffness and strength using one, two or three transverse bulkheads. These were, denoted concept A, B and C, respectively (see
Figure 3). Concept A uses one supporting bulkhead across the hull, with a height equivalent to the sides. Concept B uses two bulkheads with varying heights that extend above and over the casing. Finally, concept C has an additional bulkhead placed towards the rear of the Optimist. In all concepts there is an additional bulkhead placed in the front of the Optimist. This will be used to secure the mast. To save weight, the casing and the bulkheads were designed as sandwich structures, with the same material as for the foiling components.
The complexity of the geometry did not allow for meaningful handbook calculations of the casing or stiffening bulkheads. Instead, different lay-ups for the bulkheads and the casing, respectively, were analysed using FE. In the FE models, the casing and the bulkheads were modelled using layered shell elements only, whereas the hull was modelled with a combination of solid elements for the core and layered shell elements for the face sheets.
Fibre reinforcements oriented along the bulkhead and the casing (oriented along the heave axis) are crucial for sufficient bending stiffness and strength. Furthermore, reinforcements oriented in ±45° to these respective directions are required to carry torsional loads and reinforcements oriented at 90° are necessary to carry unexpected loads. After studying a number of alternative designs, the final lay-ups, for which results are presented below, were obtained as [02/−45/90/45/0/−45/0/45/0/core]S for the bulkheads and [90/−45/0/45/0]S for the casing. For the bulkheads, a core thickness of 12 mm was used. This yields an estimated weight increase for the three concepts of 4.0 kg for concept A, 3.9 kg for concept B and 3.8 kg for concept C, all below the required maximum.
To determine the torsional stiffness of the hull for each concept, the front of the FE model of the Optimist was constrained with a fixed support (blue surface) at the same time as a moment
M (red arrow) was applied to the aft, see
Figure 4a. The resulting rotational deformations,
, around the surge axis of the Optimist were measured at the point of moment application. The rotational stiffness,
k, was then calculated as
k = M/.
A conservative approach was taken in the strength analysis, by defining a fixed boundary on the rim of the Optimist, see
Figure 4b where the blue area on the rim represents the fixed support and where the red regions correspond to areas with applied loads. Furthermore, as the centreboard was designed in a separate part of the project, it was here replaced by a steel component with the same geometry, resulting in an even more conservative load case. The contact between the centreboard and its casing was defined such that the centreboard was free to displace within the casing but was restricted to move in the vertical direction. Furthermore, the top of the casing and the centreboard was set as a bonded connection.