From bending moment to laminate

A simple and crude way to calculate the laminate thickness of the mast is to assume the mast is an I-beam.

For the graph below I also used the following assumptions:

-Righting moment of the yacht is 60 kNm (see previous posts);

-The mast is an I-beam with 100 mm wide flanges;

-A safe maximum stress (compression/tension) in the carbon laminate is 200 N/mm2.

The graph shows the carbon laminate should be more than 10 mm thick around the upper bearing ( 0 m ).

Mast: first calculations

The freestanding mast is only supported at deck level and at the bottom of the hull. The distance between the deck bearing and the bottom bearing is around 2 metres. The distance from the deck to the top of the mast is around 17 metres.

To estimate the forces and bending moments I will first assume the mast must be able to deal with a righting moment of 60 kNm.

I will use two methods/models. Model A puts the full force of the wind at the top of the mast. Model B is more realistic and assumes the wind force is distributed evenly along the entire mast (above deck ofcourse).

The drawings below show the forces in equilibrium. Note that the maximum bending moment (60 kNm at deck level) and the load on the bottom bearing (60 kNm / 2 m = 30 kN) do not depend on the model. These are determined by the 60 kNm righting moment of the yacht.

Also note that model B allows a higher wind force (17 m x 0.42 kN/m = 7 kN) than model A (3.5 kN) and, as a result, causes a higher load on the deck bearing (37 kN versus 33.5 kN).

To calculate an efficient (not too heavy and expensive) carbon laminate of the mast at different locations I need more bending moments than the maximum of 60 kNm at deck level.

Luckily, using model A, these bending moments are very easy to calculate: the bending moments decrease linearly from 60 kNm at deck level to zero towards the bottom and the top of the mast. Using model B the calculations are more complicated (I will probably add two bending moment diagrams later to illustrate this point).

However, because the (above deck) bending moments in model B are equal to or lower than in model A, I can also choose to just ignore model B. Using model A will automatically create a safety factor AND make my life easier.

Model A: 3.5 kN point load at top of mast

Model B: 0.42 kN/m distributed load

Keel floors and bolts: first calculations

Keel floors are lateral structural beams on the inside of the hull. They strengthen the area where the keel meets the hull. This heavily loaded area also needs longitudinal strengthening/stiffening. I will deal with that later, this is about the floors and bolts.

The structural grid will probably look something like this (top view):

To determine the required strength of the floors and keel bolts two extreme scenarios must be examined: a 90 degrees knockdown and a grounding at speed.

90 degrees knockdown:

The centre of gravity of the keel is around 2 metres below the hull (I will err on the side of caution and use round numbers). The associated moment is M = 30 kN x 2 m = 60 kNm.

This moment is transferred to two rows of keel bolts on the keel flange. The distance between the rows is (almost) 0.4 m. This causes a force F = 60 kNm / 0.4 m = 150 kN on each row. Or to be precise: the bolts on one side of the flange are under tension. On the other side the edge of the flange is pushed against the hull.

There are 5 bolts per row. The force per keel bolt therefore is 150 kN / 5 = 30 kN. Galvanized steel keel bolts with size M30 8.8 (which should be able to easily take 140 kN, after a 60% preload) will be more than adequate. The safety factor is 140 kN / 30 kN, so almost 5.

The forces on the keel bolts and opposing edge are transferred to floors on the inside of the hull. This causes a bending moment in these floors.

For now the drawing below assumes there will only be one floor. The maximum bending moment in this floor depends on floor length. If the keel floor is 2 metres long, the reaction forces on the right and left edges are F = 60 kNm / 2 m = 30 kN. The resulting maximum bending moment is located at the arrows and will be M = 30 kN x 0.8 m = 24 kNm. This bending moment reduces linearly to zero at the edges of the beam/floor.

If the keel floor is 1 metre long the reaction forces at the edges will be F = 60 kNm / 1 m = 60 kN and the corresponding maximum bending moment will be M = 60 kN x 0.3 m = 18 kNm:

These bending moments determine the geometry and laminate thickness of the keel floors. Note again that the drawing above assumes there is only one floor. In reality there will be more. If there are five floors the bending moments mentioned above reduce to 24 kNm / 5 = 4.8 kNm and 18 kNm / 5 = 3.6 kNm.

The 90 degrees knockdown scenario, however, is almost irrelevant because floors and bolts that can survive a grounding at speed will be more than strong enough to survive a knockdown. As will be shown below.

Grounding at speed:

The book Principles of Yacht Design assumes that a dramatic grounding causes a decelaration of 16.5 m / s². This deceleration (e.g. from 8 knots to a complete stop in just 0.25 seconds!) will be used to calculate the force on the yacht during a grounding. I hope my future groundings will be less violent!

The absolute maximum total mass of the yacht will be around 8000 kg. The force on the yacht during a grounding is F = 8000 x 16.5 = 130 kN.

The distance between the bottom of the keel (where the yacht hits the ground) and the flange on the top of the keel is around 2 times the length of the flange. The upward and downward forces on the keel flange must therefore be F = 2 x 130 kN = 260 kN:

Theze 260 kN point loads cause very high bending moments in the floors located at the front and the back of the keel flange (keel bolts 1+2 and 9+10 respectively). The resulting bending moment depends on the unsupported length of the floors (determined by the distance between the two longitudinal structural beams, see first drawing). If, for example, the unsupported length is 1 m the maximum bending moment in the floors is M = (260 kN / 2) x 0.5 m = 65 kNm. If the unsupported length is 0.6 m the bending moment is M = (260 kN / 2) x 0.3 = 39 kNm.

These bending moment values on the two floors at the front and the back are (much) higher than calculated above for a 90 degrees knockdown:

The fact that the 260 kN point loads should ideally be distributed across the entire 0.4 m width of the keel flange (under compression at the back) or be divided equally between the two bolts (under tension at the front) improves this situation a little. The maximum bending moments decrease, especially at the front (bottom drawing):

The last drawing can also be used to check the size of the (forward) two keel bolts. During a grounding each keel bolt must be able to handle 130 kN. This means there is no real safety factor left for the M30 8.8 bolts mentioned earlier: 130 kN is almost equal to 140 kN.

Principles of Yacht Design, however, states that it is reasonable to include all the bolts in the forward 25% of the keel flange for this calculation. There are 4 bolts in this area: 260 kN / 4 = 65 kN per bolt. That would improve the safety factor to more than 2 (140 kN / 65 kN).

If the assumed 60% preload of the bolts (used to determine the 140 kN) would be reduced a bit, the safety factor for the bolts would become acceptable again. Right now M30 8.8 bolts still seem strong enough, even at the front of the keel flange.

Leveling the mast mold

Leveling the mold (half) with a laser.

Conclusion: the concrete floor was not particularly level!

Trying to determine an acceptable transition from round (bottom 2 metres of the mast) to wing shape (the rest).

Do not like this. The shape should change a little more gradually. Never mind, will try again.