Static Behavior (1) - Homogeneous Rode

1. Rode Requirements

In the tackle chapter, we listed the qualities required from an anchoring rode:

• Req. 4: The rode should pull the anchor parallel to the bottom.
• Req. 5: The rode should be easy to stow, wind and unwind aboard.
• Req. 6: The rode should reduce the tensile stresses caused by gusts/waves on the anchor and boat accessories.

In this chapter, we deal with Req. 4 and 5 (Req. 6 will be studied in the dynamic behavior chapter).

Obviously, rigid devices (e.g. a steel bar) do not meet these requirements! The latter also eliminates cables. Other candidate materials are: chain, textile rope (e.g. nylon), composite ropes, or some combination of these. In most cases, such materials are homogeneous, i.e. their physical characteristics are constant along their whole length (weight per unit of length, elasticity, breaking load etc.).

2. Theoretical Catenary

If you hang some length of an homogeneous, non-elastic and perfectly flexible rode between 2 fixed points, its equilibrium figure is a segment of "catenary". This famous mathematical curve (its equation was established 3 centuries ago!) looks like a parabola, but its lower part is slightly flatter (fig. 1.1).

Figure 1.1 - Catenary between 2 fixed points

3. Actual Homogeneous Rode Figure

An anchoring situation is somewhat different of the one above:

• One of the hanging points (B at the bow roller) is mobile and pulled by a horizontal force F due to the wind pressure against the boat.
• The second point (A at the anchor pulling eye) is fixed on the bottom, so that the rode cannot hang below it (fig. 1.2).

Figure 1.2 - Typical figure of an anchoring rode

To keep things simple, at this stage, we make several assumptions (discussed elsewhere in these pages):

• The anchor has been correctly set (see the anchor chapter),
• The rode is located in a vertical plane (R is the swinging radius),
• Its stretch is negligible,
• Its friction on the bottom is negligible,
• Neither the boat nor the rode have inertia,
• The pulling force F is constant and located inside the rode plane.
• The system has reached its equilibrium point, i.e.the force F due to the wind pressure is exactly compensated by an opposite force due to the weight of the rode.

Thanks to the gravity, the rode acts as a "pseudo-spring", even though we have just assumed it has no elasticity of its own (which is false for real nylon rode, but almost true for steel chain - more on that in next pages).

Obviously, the active (i.e. veered out) rode length L must be greater than the height H of the roller above the bottom, otherwise the anchor would not even reach it! By the way, please note that H = water depth + freeboard at the boat's bow. In shallow anchorages, this can be significant: for example, if the water depth is 2 m (6 ft) and the freeboard is 1 m (3 ft), the effective height H is 3 m (9 ft)!

In these conditions, the rode figure varies according to 4 parameters:

• The height H,
• The active rode length L,
• The rode weight (in the water) per unit of length w,
• The pulling force F.

An example using a chain rode:

For H = 5 m (16 ft) and L = 15 m (49 ft) of 11 mm (7/16") chain, which weighs approximately w = 2.5 daN/m (1.7 lb/ft) in seawater, let's see what successively happens when F varies from 0 up to a very high value (fig.. 1.3):

Figure 1.3 - Homogeneous rode profiles for various pulling forces

• 1. For F = 0 (gray curve), the rode is L-shaped: the upper part hangs vertically from the bow down to the bottom, the rest lies on the bottom right down to the anchor.
• 2. When F increases (10 daN - 22 lb - green curve), the boat moves backwards, so the lying part of the rode lifts up progressively. The shape of the part between the bow B and the contact point C with the bottom is a catenary. Anyway, the tension on the pulling eye remains horizontal and equal to F, so the anchor shank remains horizontal too, and the holding power is unchanged.
• 3. When F reaches a critical value F_c (52 daN - 116 lb - orange curve), the whole rode is lifted above the bottom, i.e. C has merged with A. The rode is tangent to the bottom at the shank pulling eye, the holding power is still unchanged.
• 4. Beyond F_c (e.g. 100 daN - 220 lb - red curve), the tension on the pulling eye is no longer horizontal. The shank lifts up, and the holding power begins to decrease (see the anchor chapter).
• 5. If F continues to increase, the catenary reduces to an almost straight line (black curve), and eventually the anchor pulls out.

What if we use nylon instead of chain?

With a 22 mm (7/8") nylon rode of same length, the critical value F_c drops down to 0.5 daN (1.2 lb)! This is due to the 90% apparent weight loss of the nylon in seawater. Even with a 45 m (150 ft) line, F_c reaches a mere 6 daN (12 lb). This explains that, in practice, it is impossible to keep the anchor flat on the bottom with an all-nylon rode, as soon as the wind exceeds a few knots.

4. Fundamental Homogeneous Rode Equations

Given w and the height H, a pulling force F lifts rode length L_up:

 (1.1)

L_up is the minimum rode length to keep the shank horizontal for a pulling force F.

Conversely, the critical force F_c that lifts the whole rode (length L) is:

 (1.2)

F_c is the maximum pulling force to keep the shank horizontal for a rode length L.

F_c is the main factor that affects security. It depends on the 3 parameters H, L, w as follows:

From this formula, we can notice interesting properties:

• 1. The critical force F_c is proportional to w. Consequently, for a given length L, using e.g. a 10 mm (3/8") instead of 8 mm (5/16") chain increases F_c by 44%. The main drawback is a 44% increase of the weight carried on board.
• 2. F_c is approximately proportional to the square of length L. Consequently, increasing L by e.g. 41% multiplies F_c by 2. The main drawbacks are a 41% increase of the weight carried on board, and a 41% increase of the swinging radius.

We can rewrite expression(1.1) to compute the minimum scope N (ratio L/H) to keep the shank horizontal for a pulling force F:

    (1.3)

This equation shows that, contrary to a popular belief, the minimum scope depends on both parameters F and H. An example will be more convincing:

Let's take a 8 mm (5/16") chain. What is the minimum scope to withstand F = 89 daN (200 lb)?

If the pulling force F exceeds F_c, the rode is no longer tangent to the bottom at the shank pulling eye: its angle , sometimes called angulation (see fig. 1.4a) can be computed from the following function:

    (1.5)

This expression allows computing the pulling force given an accepted angulation (anchors are more or less tolerant to a moderate angulation, at the cost of reduced holding power - see discussion in the anchor chapter). One can check that F(0) = F_c.

In the all-chain example of fig. 1.3, allowing a 6 degrees angulation instead of 0 would increase the acceptable force by 46%.

With an all-textile rode, angulation is practically unavoidable, and keeping it at an acceptable value needs a very large scope. As the shape of the rode is very close to a straight line (fig. 1.4b) its slope tan is virtually equal to the scope N = L/H. In this case, whatever the (high) pulling force, the traditional 10:1 scope leads to a 5.7 degrees angulation (which is acceptable for some modern anchors - see the anchor chapter).

Figure 1.4a - Angulation		Figure 1.4b - All-textile rode (high pulling force case)

6. Pulling Force vs. Boat Drift

As we shall see in the dynamic behavior chapter, knowing the static relation between the pulling force and the horizontal coordinate of the boat is of prime importance to predict the movement of the boat and the strains that will affect the anchoring tackle under wind gusts. For the all-chain rode of fig. 1.3 example, this relation is charted on fig. 1.4:

Figure 1.5 - Boat Drift vs. Pulling Force

A striking characteristics of this relation is its high non-linearity: a given force variation causes much less drift at high pulling forces than at low ones. When the rode tightens, the pseudo-elasticity due to gravity vanishes, so the force necessary to move the boat back increases asymptotically.

7. Simulate Your Own Homogeneous Rode

To go into the above subjects in greater depth, to try other cases, to compare various situations etc., you can open (or download) the spreadsheet sta_hom.xls (requires Microsoft® Excel 2011 or +).

8. Conclusions

Let's check both technologies against the criteria defined at the top of this page (Req. 4 and 5):

Thus, except for moderate weather conditions, none of these homogeneous technologies are really satisfactory. In the next page, we shall study the static behavior of heterogeneous rodes.

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