Dynamic Stability Analysis of Sail Ship Masts

A reliable mast is critical to the safety and performance of a sail ship.  The collapse of a mast not only impairs the mobility of a ship, but can cause injuries to the passengers and crew, and also serious damage to the ship.

Engineers designing a mast need to conduct comprehensive analyses of the mast under different loading cases. Finite element analysis has proved to be efficient for this purpose. In fact, some ship classification societies, such as the Germanischer Lloyd (DNV GL) in Germany, prescribe a finite element analysis in their design guidelines for the classification of large sail ship masts and rigging. In this Brief, we feature the use of ADINA for designing safe and reliable masts for large modern sail ships, see also ref.

The modern sail ship mast is typically a hollow column made of aluminum alloy. Figure 1 shows the cross section of an actual aluminum alloy mast used in the yacht industry.

Figure 1  Mast cross section, about 900mm by 400mm

The total length of the mast of a large sail ship can be 60 or 70m, about the same order of magnitude as the length of the ship, but typically the unsupported span length of a bottom mast panel (the lower span of the mast) is approximately 10m. Figure 2 shows the ADINA finite element model. The bottom of the mast is fully clamped. A compression loading of 5000kgf is applied on the top. The loading is used to represent the weight of the upper rigs and the pre-tension introduced by the mast tuning.

Figure 2  The finite element model of the mast, about 10m long

A preliminary quasi-static collapse analysis is first performed to identify the post-buckling response of the mast. The computation consists of a linearized buckling mode step and a static step. In the linearized buckling mode step ADINA extracts the buckling shapes of the masts (the eigenmodes). In the static step an assumed initial imperfection is applied based on the buckling shapes. The load-displacement curve then calculated indicates the progressive collapse behavior, as shown in Figure 3. The progressive collapse is clearly a global buckling phenomenon, as can be seen in the post-buckling shape of the buckled mast (Figure 4).

Figure 3  Progressive collapse analysis: compressive force vs. longitudinal displacements in the z-direction, showing pre- and post-buckling stages up to collapse

Figure 4  Mast collapse due to axial static compressive loading, showing band plot of von Mises effective stress

A more complex instability issue is the collapse under impulse dynamic loads. The dynamic load comes from multiple sources, including wind gusts, sudden course variations, and abrupt accelerations/decelerations. Such dynamic loads can well exceed the load limit established in the quasi-static analysis, but usually act on the mast for only a short period of time.

For simplicity, the dynamic load is defined as a half-sinusoidal impulse, see Figure 5. The load factor is the ratio of the dynamic load to the quasi-static stable load limit. Different wavelengths and factor amplitudes are employed in a series of dynamics analyses to establish the mast response (see Figure 6).

Figure 5  Time loading functions adopted in dynamic analysis

Figure 6  Dynamic response chart of the mast

As seen in Figure 6, the stability of the mast depends not only on the load magnitude, but also on the period of the load. Another observation is that although under quasi-static loading the mast always experiences a global instability (collapse in the first eigenmode), it can experience local instability when it is subjected to a dynamic load. The above movie shows the collapse of the mast under dynamic loading.

It is seen that the comprehensive range of analysis capabilities in ADINA is an asset in the design and engineering of structures pertaining to ships and boats.

Reference

• M. Gaiotti and C. M. Rizzo, "Dynamic buckling of masts of large sail ships", Ships and Offshore Structures, 2014.

Keywords:
Mast, sail ship, ship structure, buckling, frequency, eigenmode, quasi-static, dynamic analysis

Courtesy of M. Gaiotti and C. M. Rizzo, University of Genova, Italy