## Analysis of the factor loadings

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- Category: yacht construction

It should be underlined that, in the last decade, significant progress has been made in the field of aerodynamic and hydrodynamic load analysis, thanks to some prestigious international competition such as the America's Cup and Admiral's Cup. Many problems related to the forces generated by sails have been faced by developing powerful numerical software able to take into account a large number of geometric and physical variables. Nevertheless some problems remain like, as an example, the behaviour of rigging under wind gust, the dynamic loading on the mast caused by the boat movements in rough sea and the pretensioning loads induced by initial rigging trim.

These aspects are currently under investigation but, up to now, it is difficult to have reliable results. The consequence of these, and other, uncertainties is that the range of safety coefficients becomes very wide and it is very difficult to choose the correct one. Too much severe loads will result in very safe but low performances rigging, too much optimistic loads will result in good performances but unreliable rigging. The correct choice should individuate a compromise able to obtain a good rigging system with "reasonably good performances" and "reasonably safe configuration". This choice does not depend only on technical aspects but also on the skill of the crew: it is obvious that the safety level of a racing yacht cannot be the same of a cruise one. A review of methods used for the calculation of mast and rigging design loads is presented in the following, covering the development from the very old empirical ones, up to present numerical methodologies. The loads applied to the mast are mainly due to the forces developed by sails, by pretensioning loads and by the hydraulic jack used to raise the mast. An approximate evaluation can be performed on the basis of a uniform pressure distribution as a function of weather conditions and wind speed (Marchaj, 1979). A more rigorous approach, although roughly simplified, consists in considering the equilibrium between hydrodynamic forces acting on the hull and the aerodynamic forces developed by the sails. The amount of the propulsive effect generated by sails depends on a number of factors such as: the area and geometry of sails, the apparent wind velocity and the angle of incidence of sails. The resultant of sails forces FT can be decomposed into lift L (normal to the apparent wind direction) and drag D (opposite to the apparent wind direction) and expressed in terms of non-dimensional coefficients CL and CD. Lift and drag can be measured in the wind tunnel during experimental tests on scale models and reported in polar diagrams as a function of the angle of incidence α.

The total force FT can also be decomposed into two other components: the driving force FR in the direction of the boat's course and the heeling force FH perpendicular to the boat's course; also in this case non-dimensional coefficients CR and CH are defined. To compute the load exerted by sails on the mast it is then necessary to know the coefficients CL and CD or CR and CH; a great deal of experimental data on sails has been collected by researchers in wind tunnel tests and some of them are available in literature, such as those published by Marchaj (1962, 1964) or the data collection gathered on board Bay Bea yacht (Kerwin et. al., 1974). The sail forces FR and FH can also be determined by considering the hydrostatic properties of the hull in heeled conditions. The heeling moment MH caused by the action of the wind on sails is balanced by the righting moment MR rising when the boat heels. The righting moment for an angle of heel θ is equal to Δ×GZθ where Δ is the displacement of the yacht and GZθ the righting arm. The side force FH can be determined as follows:

where h is the vertical distance between sails' centre of effort (aerodynamic) and hull centre of lateral resistance (hydrodynamic). From the cross curves of the hull it is possible to know exactly the force necessary to heel the yacht of an angle θ; this will be the transverse force developed by the sails in a quasi-static condition. Assuming proper sail coefficients at the design heel angle θ, the apparent wind velocity and the driving force FR can be determined. The starting point for the designer is then to determine the maximum heel angle θ to be assumed for the calculation. For little and medium size sailboats the reference heel angle for mast and rigging scantling is typically 30°. In the case of big sailing yachts this could be too large and might lead to excessive mast section dimensions; thus a maximum heel of 20-25° is often assumed. Once the driving and heeling forces FR and FH have been calculated and subdivided between mainsail and foresail, the next problem to solve is how those forces should be applied on mast and rigging. In a simplified approach it can be assumed that the mainsail transmits to the mast a distributed load along its length. The simplest way to apply this load is by a triangular shape as shown in Figure 13a. Taking into account that the pressure on the upper part of the sail is greater, owing to the higher wind velocity, a trapezoidal distribution (Figure 13b) would be more suitable. According to lifting line theory the pressure follows an elliptical distribution because of the vortex rising at the upper and lower sail bounds (Figure 13c). The actual pressure distribution will vary dynamically depending on aspect ratio, twist and sheet tension. For application to a finite element model, this type of distribution can be well approximated by a step distribution as shown in Figure 13d. Such a distribution of the load is conservative towards the bending moment on the mast because the centre of application of the resulting force is higher than other ones.

Figure 13: Distribution of mainsail load on mast: (a) triangular; (b) trapezoidal; (c) elliptical; (d) step varied (Claughton et.al., 1998).

As far as the foresail is concerned the total force can be split between the forestay and the jibsheet. The percentage depends on the tension in the halyard but it is reasonable to consider that 20% is supported by the sheet and the 80% by the jibstay and, consequently charged on the masthead. The force on the masthead depends on the tension in the jibstay and it is a function of the maximum jibstay deflection. The jibstay tension can be estimated considering the deformed shape of the stay to be a very tight catenary supporting a distributed load along its length. To know the tension in the jibstay, it is necessary to impose a minimum, reasonable value for the maximum deflection. It is sometimes argued that the curvature of the forestay could cause a stagnation effect on the mainsail and thus consequently decrease the propulsive force component FR. In order to reduce this effect it is a common practice to pretension the stay as much as possible increasing the compression and bending stresses on the mast. In current practice, it can be assumed a maximum stay deflection between 2 and 5% of the jibstay length. There are other loads to consider such as those transmitted by boom, the compression at mast step by an hydraulic jack, the pretensioning of stays and shrouds and the tension of halyard. In the case of a linear analysis maximum values of considered loads should be applied to the model. The results of the calculation will be analysed in terms of stresses and displacements.

For what the displacements are concerned it is a common practice not to allow displacements at the top of the mast higher than 2% of the total mast height with garage storage. To calculate the pressure field developed by the wind on sails of different shapes in upwind condition, even interacting each other, numerical methods based on lifting line or lifting surface theories (Milgram 1968, Greeley et.al. 1989, Jackson 1984) are now available. Nowadays modern methodologies allow designers to perform more sophisticated analyses. Using CFD codes (Computational Fluid Dynamics) it is possible to estimate the flow around the mast and the interaction between sails and standing rigging, obtaining more precise information about loads. It is essential to model the totality of the fluid-structure interaction in order to accurately determine the loads. Nevertheless all described methods require validations by experimental or real scale measurements. Some researchers in the past instrumented sailing yachts to collect load data on mast, shrouds and halyards (Enlund et.al., 1984). More recently, Hansen et.al. (2007) presented an investigation on sail forces measurement in wind tunnel and on board a full scale sailing boat. These days some shipyards producing very large sailing yachts feel the necessity to achieve a more detailed knowledge of loads acting on the masts and rigging by implementing permanent instrumentation on their ships. A significant example here is represented by the instrumentation installed on the Perini schooner "Maltese Falcon" and described by Roberts and Dijkstra (2004). The same activity is under course in the Department of Naval Architecture of the University of Genova where a new measurement system has been realized and it will be installed on another Perini Navi sailing yacht (Rizzo & Carrera, 2006).